2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 6.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.4% 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, 0.5× 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 76.5%

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

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-304)
   (* (pow (- a) -0.3333333333333333) (cbrt (* g -0.5)))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-304) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((g * -0.5));
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-304) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((g * -0.5));
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -4e-304)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(g * -0.5)));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -4e-304], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g * 0.5), $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) < -3.99999999999999988e-304

   1. Initial program 78.4%

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

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. cbrt-prod N/A

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

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

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

      10. times-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cbrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-cbrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. div-inv N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-cbrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-cbrt.f64 N/A

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

      12. cbrt-div N/A

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

      13. un-div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. associate-*r/ N/A

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

      19. div-inv N/A

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

      20. distribute-neg-frac2 N/A

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

      21. lift-/.f64 N/A

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

      22. *-commutative N/A

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

   6. Applied rewrites 91.8%

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

   if -3.99999999999999988e-304 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.8%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.3

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

   4. Applied rewrites 92.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\frac{2}{\frac{-1}{a}}}{-g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 2e-302)
   (/ 1.0 (cbrt (/ (/ 2.0 (/ -1.0 a)) (- g))))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-302) {
		tmp = 1.0 / cbrt(((2.0 / (-1.0 / a)) / -g));
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-302) {
		tmp = 1.0 / Math.cbrt(((2.0 / (-1.0 / a)) / -g));
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 2e-302)
		tmp = Float64(1.0 / cbrt(Float64(Float64(2.0 / Float64(-1.0 / a)) / Float64(-g))));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 2e-302], N[(1.0 / N[Power[N[(N[(2.0 / N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] / (-g)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g * 0.5), $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) < 1.9999999999999999e-302

   1. Initial program 78.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. pow2 N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied rewrites 35.5%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4}} \cdot \left(2 \cdot a\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{\left(a \cdot a\right) \cdot 4}}} \]

      5. clear-num N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      6. cbrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{g \cdot \left(2 \cdot a\right)}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(2 \cdot a\right)}}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(a \cdot 2\right)}}}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{\color{blue}{\left(g \cdot a\right) \cdot 2}}}} \]

      14. times-frac N/A

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a \cdot a}{g \cdot a} \cdot \frac{4}{2}}}} \]

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 33.4

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

   6. Applied rewrites 33.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/r/ N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/r/ N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. frac-2neg N/A

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

      21. lower-/.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 1.9999999999999999e-302 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.6%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.3

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

   4. Applied rewrites 92.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\frac{2}{\frac{-1}{a}}}{-g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 5: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\frac{2}{\frac{-1}{a}}}{-g}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{g \cdot 0.5}{\sqrt[3]{a \cdot \left(\left(g \cdot 0.5\right) \cdot \left(g \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{2}{\frac{g}{a}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))))
   (if (<= t_0 -5e-321)
     (/ 1.0 (cbrt (/ (/ 2.0 (/ -1.0 a)) (- g))))
     (if (<= t_0 2e-322)
       (/ (* g 0.5) (cbrt (* a (* (* g 0.5) (* g 0.5)))))
       (/ 1.0 (cbrt (/ 2.0 (/ g a))))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-321) {
		tmp = 1.0 / cbrt(((2.0 / (-1.0 / a)) / -g));
	} else if (t_0 <= 2e-322) {
		tmp = (g * 0.5) / cbrt((a * ((g * 0.5) * (g * 0.5))));
	} else {
		tmp = 1.0 / cbrt((2.0 / (g / a)));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-321) {
		tmp = 1.0 / Math.cbrt(((2.0 / (-1.0 / a)) / -g));
	} else if (t_0 <= 2e-322) {
		tmp = (g * 0.5) / Math.cbrt((a * ((g * 0.5) * (g * 0.5))));
	} else {
		tmp = 1.0 / Math.cbrt((2.0 / (g / a)));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-321)
		tmp = Float64(1.0 / cbrt(Float64(Float64(2.0 / Float64(-1.0 / a)) / Float64(-g))));
	elseif (t_0 <= 2e-322)
		tmp = Float64(Float64(g * 0.5) / cbrt(Float64(a * Float64(Float64(g * 0.5) * Float64(g * 0.5)))));
	else
		tmp = Float64(1.0 / cbrt(Float64(2.0 / Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-321], N[(1.0 / N[Power[N[(N[(2.0 / N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] / (-g)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-322], N[(N[(g * 0.5), $MachinePrecision] / N[Power[N[(a * N[(N[(g * 0.5), $MachinePrecision] * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(2.0 / N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.99994e-321

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. pow2 N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied rewrites 43.2%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4}} \cdot \left(2 \cdot a\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{\left(a \cdot a\right) \cdot 4}}} \]

      5. clear-num N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      6. cbrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{g \cdot \left(2 \cdot a\right)}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(2 \cdot a\right)}}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(a \cdot 2\right)}}}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{\color{blue}{\left(g \cdot a\right) \cdot 2}}}} \]

      14. times-frac N/A

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a \cdot a}{g \cdot a} \cdot \frac{4}{2}}}} \]

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 39.5

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

   6. Applied rewrites 39.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/r/ N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/r/ N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. frac-2neg N/A

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

      21. lower-/.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if -4.99994e-321 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.97626e-322

   1. Initial program 5.7%

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

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. cbrt-prod N/A

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

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

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

      10. times-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cbrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-cbrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. div-inv N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \sqrt[3]{g \cdot \frac{1}{2}}\right)} \cdot \frac{-1}{\sqrt[3]{a}} \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)}} \cdot \frac{-1}{\sqrt[3]{a}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{{0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)} \cdot \color{blue}{\frac{-1}{\sqrt[3]{a}}} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}\right) \cdot -1}{\left(0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)\right) \cdot \sqrt[3]{a}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}\right) \cdot -1}{\left(0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)\right) \cdot \sqrt[3]{a}}} \]

   6. Applied rewrites 24.9%

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

   if 1.97626e-322 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. pow2 N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied rewrites 50.6%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4}} \cdot \left(2 \cdot a\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{\left(a \cdot a\right) \cdot 4}}} \]

      5. clear-num N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      6. cbrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{g \cdot \left(2 \cdot a\right)}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(2 \cdot a\right)}}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(a \cdot 2\right)}}}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{\color{blue}{\left(g \cdot a\right) \cdot 2}}}} \]

      14. times-frac N/A

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a \cdot a}{g \cdot a} \cdot \frac{4}{2}}}} \]

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 45.0

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

   6. Applied rewrites 45.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 91.5

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

   8. Applied rewrites 91.5%

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{2}{\frac{g}{a}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\frac{2}{\frac{-1}{a}}}{-g}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{g \cdot 0.5}{\sqrt[3]{a \cdot \left(\left(g \cdot 0.5\right) \cdot \left(g \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{2}{\frac{g}{a}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{g \cdot 0.5}{\sqrt[3]{a \cdot \left(\left(g \cdot 0.5\right) \cdot \left(g \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{2}{\frac{g}{a}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))))
   (if (<= t_0 -5e-321)
     (/ 1.0 (cbrt (/ a (* g 0.5))))
     (if (<= t_0 2e-322)
       (/ (* g 0.5) (cbrt (* a (* (* g 0.5) (* g 0.5)))))
       (/ 1.0 (cbrt (/ 2.0 (/ g a))))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-321) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_0 <= 2e-322) {
		tmp = (g * 0.5) / cbrt((a * ((g * 0.5) * (g * 0.5))));
	} else {
		tmp = 1.0 / cbrt((2.0 / (g / a)));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-321) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_0 <= 2e-322) {
		tmp = (g * 0.5) / Math.cbrt((a * ((g * 0.5) * (g * 0.5))));
	} else {
		tmp = 1.0 / Math.cbrt((2.0 / (g / a)));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-321)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_0 <= 2e-322)
		tmp = Float64(Float64(g * 0.5) / cbrt(Float64(a * Float64(Float64(g * 0.5) * Float64(g * 0.5)))));
	else
		tmp = Float64(1.0 / cbrt(Float64(2.0 / Float64(g / a))));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-321], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-322], N[(N[(g * 0.5), $MachinePrecision] / N[Power[N[(a * N[(N[(g * 0.5), $MachinePrecision] * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(2.0 / N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.99994e-321

   1. Initial program 88.5%

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

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 89.6

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

   4. Applied rewrites 89.6%

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

   if -4.99994e-321 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.97626e-322

   1. Initial program 5.7%

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

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. cbrt-prod N/A

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

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

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

      10. times-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cbrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-cbrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. div-inv N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \sqrt[3]{g \cdot \frac{1}{2}}\right)} \cdot \frac{-1}{\sqrt[3]{a}} \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)}} \cdot \frac{-1}{\sqrt[3]{a}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{{0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)} \cdot \color{blue}{\frac{-1}{\sqrt[3]{a}}} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}\right) \cdot -1}{\left(0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)\right) \cdot \sqrt[3]{a}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {\left(\sqrt[3]{g \cdot \frac{1}{2}}\right)}^{3}\right) \cdot -1}{\left(0 \cdot 0 + \left(\sqrt[3]{g \cdot \frac{1}{2}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} + 0 \cdot \sqrt[3]{g \cdot \frac{1}{2}}\right)\right) \cdot \sqrt[3]{a}}} \]

   6. Applied rewrites 24.9%

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

   if 1.97626e-322 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. pow2 N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied rewrites 50.6%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4}} \cdot \left(2 \cdot a\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{\left(a \cdot a\right) \cdot 4}}} \]

      5. clear-num N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      6. cbrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{g \cdot \left(2 \cdot a\right)}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(2 \cdot a\right)}}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(a \cdot 2\right)}}}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{\color{blue}{\left(g \cdot a\right) \cdot 2}}}} \]

      14. times-frac N/A

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a \cdot a}{g \cdot a} \cdot \frac{4}{2}}}} \]

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 45.0

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

   6. Applied rewrites 45.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 91.5

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

   8. Applied rewrites 91.5%

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{2}{\frac{g}{a}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{g \cdot 0.5}{\sqrt[3]{a \cdot \left(\left(g \cdot 0.5\right) \cdot \left(g \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{2}{\frac{g}{a}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. clear-num N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

   13. div-inv N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 76.8

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

4. Applied rewrites 76.8%

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

Alternative 8: 76.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. sqr-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. pow2 N/A

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

   11. remove-double-neg N/A

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

   12. remove-double-neg N/A

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

   13. sqr-neg N/A

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

   14. pow-prod-down N/A

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

   15. pow-prod-up N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

4. Applied rewrites 40.2%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]
5. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4} \cdot \left(2 \cdot a\right)}} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(a \cdot a\right) \cdot 4}} \cdot \left(2 \cdot a\right)} \]

   4. associate-*l/ N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{\left(a \cdot a\right) \cdot 4}}} \]

   5. clear-num N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

   6. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

   9. lower-cbrt.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \left(2 \cdot a\right)}}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{g \cdot \left(2 \cdot a\right)}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(2 \cdot a\right)}}}} \]

   12. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{g \cdot \color{blue}{\left(a \cdot 2\right)}}}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{\color{blue}{\left(g \cdot a\right) \cdot 2}}}} \]

   14. times-frac N/A

       \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a \cdot a}{g \cdot a} \cdot \frac{4}{2}}}} \]

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 36.3

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

6. Applied rewrites 36.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. *-inverses N/A

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

   10. metadata-eval N/A

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

   11. div-inv N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. associate-*l/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 76.8

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

8. Applied rewrites 76.8%

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

9. Final simplification 76.8%

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

Alternative 9: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

3. Final simplification 76.5%

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

Alternative 10: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 76.5

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

4. Applied rewrites 76.5%

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

5. Final simplification 76.5%

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

Reproduce

herbie shell --seed 2024231 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))