2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 4.9s

Alternatives: 8

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, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. neg-mul-1 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 98.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 98.7

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

6. Applied rewrites 98.7%

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

   1. rem-cube-cbrt N/A

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

   2. pow1/3 N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. unpow-1 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-neg.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. lower-/.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. neg-mul-1 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 98.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 98.7

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

6. Applied rewrites 98.7%

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

   1. rem-cube-cbrt N/A

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

   2. pow1/3 N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. unpow-1 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-neg.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. lower-/.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 98.7

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

8. Applied rewrites 98.7%

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 3: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -4 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot {a}^{-0.3333333333333333}\right) \cdot \sqrt[3]{g \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -4e-307)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (* (* -1.0 (pow a -0.3333333333333333)) (cbrt (* g -0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -4e-307) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = (-1.0 * pow(a, -0.3333333333333333)) * cbrt((g * -0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -4e-307) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = (-1.0 * Math.pow(a, -0.3333333333333333)) * Math.cbrt((g * -0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -4e-307)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = Float64(Float64(-1.0 * (a ^ -0.3333333333333333)) * cbrt(Float64(g * -0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -4e-307], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -3.99999999999999964e-307

   1. Initial program 76.8%

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

   4. Applied rewrites 98.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 92.3

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

   6. Applied rewrites 92.3%

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

   if -3.99999999999999964e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 70.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. cbrt-div N/A

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

      7. distribute-neg-frac N/A

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

      8. cbrt-div N/A

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

      9. pow1/3 N/A

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

      10. sqr-pow N/A

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

      11. pow-prod-down N/A

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

      12. sqr-neg N/A

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

      13. remove-double-neg N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/3 N/A

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

      18. cbrt-div N/A

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

      19. pow1/3 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 98.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

   8. Applied rewrites 92.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -4 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot {a}^{-0.3333333333333333}\right) \cdot \sqrt[3]{g \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 10^{-301}:\\ \;\;\;\;\frac{-1}{\sqrt[3]{\frac{-2 \cdot a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot {a}^{-0.3333333333333333}\right) \cdot \sqrt[3]{g \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) 1e-301)
   (/ -1.0 (cbrt (/ (* -2.0 a) g)))
   (* (* -1.0 (pow a -0.3333333333333333)) (cbrt (* g -0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 1e-301) {
		tmp = -1.0 / cbrt(((-2.0 * a) / g));
	} else {
		tmp = (-1.0 * pow(a, -0.3333333333333333)) * cbrt((g * -0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 1e-301) {
		tmp = -1.0 / Math.cbrt(((-2.0 * a) / g));
	} else {
		tmp = (-1.0 * Math.pow(a, -0.3333333333333333)) * Math.cbrt((g * -0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 1e-301)
		tmp = Float64(-1.0 / cbrt(Float64(Float64(-2.0 * a) / g)));
	else
		tmp = Float64(Float64(-1.0 * (a ^ -0.3333333333333333)) * cbrt(Float64(g * -0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 1e-301], N[(-1.0 / N[Power[N[(N[(-2.0 * a), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision] * N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 1.00000000000000007e-301

   1. Initial program 77.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. cbrt-div N/A

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

      7. distribute-neg-frac N/A

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

      8. cbrt-div N/A

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

      9. pow1/3 N/A

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

      10. sqr-pow N/A

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

      11. pow-prod-down N/A

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

      12. sqr-neg N/A

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

      13. remove-double-neg N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/3 N/A

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

      18. cbrt-div N/A

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

      19. pow1/3 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. pow1/3 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. unpow-prod-down N/A

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

      9. clear-num N/A

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

      10. lift-neg.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. unpow-prod-down N/A

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

      15. div-inv N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/r/ N/A

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

      20. unpow-prod-down N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

   8. Applied rewrites 78.0%

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

   if 1.00000000000000007e-301 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 69.8%

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

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

      6. cbrt-div N/A

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

      7. distribute-neg-frac N/A

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

      8. cbrt-div N/A

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

      9. pow1/3 N/A

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

      10. sqr-pow N/A

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

      11. pow-prod-down N/A

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

      12. sqr-neg N/A

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

      13. remove-double-neg N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/3 N/A

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

      18. cbrt-div N/A

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

      19. pow1/3 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 98.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

   8. Applied rewrites 92.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 10^{-301}:\\ \;\;\;\;\frac{-1}{\sqrt[3]{\frac{-2 \cdot a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot {a}^{-0.3333333333333333}\right) \cdot \sqrt[3]{g \cdot -0.5}\\ \end{array} \]
5. 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 73.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. pow1/3 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. unpow-prod-down N/A

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

   9. clear-num N/A

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

   10. lift-neg.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. unpow-prod-down N/A

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

   15. div-inv N/A

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

   16. associate-/l/ N/A

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

   17. lift-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. associate-/r/ N/A

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

   20. unpow-prod-down N/A

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

   21. lower-*.f64 N/A

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

6. Applied rewrites 98.7%

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

Alternative 6: 76.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. pow1/3 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. unpow-prod-down N/A

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

   9. clear-num N/A

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

   10. lift-neg.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. unpow-prod-down N/A

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

   15. div-inv N/A

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

   16. associate-/l/ N/A

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

   17. lift-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. associate-/r/ N/A

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

   20. unpow-prod-down N/A

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

   21. lower-*.f64 N/A

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

6. Applied rewrites 98.7%

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

8. Applied rewrites 73.6%

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

Alternative 7: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

Alternative 8: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow2 N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

4. Applied rewrites 73.5%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

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

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 73.5

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

6. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2024322 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))