2-ancestry mixing, zero discriminant

Percentage Accurate: 75.9% → 98.7%

Time: 5.1s

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: 75.9% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   4. associate-/l/ N/A

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

   5. cbrt-div N/A

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

   6. pow1/3 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. pow1/3 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. lower-/.f64 79.9

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

4. Applied rewrites 79.9%

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

   \[\leadsto {\left(\frac{\sqrt[3]{\frac{-1}{g}}}{\sqrt[3]{\frac{-0.5}{a}}}\right)}^{-1} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   2. metadata-eval N/A

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

   3. associate-/r/ N/A

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

   4. unpow-1 N/A

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

   5. lift-pow.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow-1 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 98.7

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

6. Applied rewrites 98.7%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   6. lift-*.f64 N/A

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

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

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

   8. lift-neg.f64 N/A

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

   9. cbrt-div N/A

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

   10. lift-neg.f64 N/A

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

   11. remove-double-neg N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lift-neg.f64 N/A

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

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

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

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

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lower-cbrt.f64 98.7

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

6. Applied rewrites 98.7%

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

Alternative 4: 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 80.7%

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

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

   6. lift-*.f64 N/A

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

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

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

   8. lift-neg.f64 N/A

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

   9. cbrt-div N/A

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

   10. lift-neg.f64 N/A

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

   11. remove-double-neg N/A

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

   12. cbrt-div N/A

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

   13. associate-*l/ N/A

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

   14. lift-/.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. neg-mul-1 N/A

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

   17. associate-*r* N/A

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

   18. cbrt-prod N/A

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

6. Applied rewrites 98.7%

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

Alternative 5: 77.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (/ g (* 2.0 a)) (- INFINITY))
   (/ (cbrt (* (* (* a a) 0.5) g)) a)
   (cbrt (/ g (/ -1.0 (/ -0.5 a))))))

C

double code(double g, double a) {
	double tmp;
	if ((g / (2.0 * a)) <= -((double) INFINITY)) {
		tmp = cbrt((((a * a) * 0.5) * g)) / a;
	} else {
		tmp = cbrt((g / (-1.0 / (-0.5 / a))));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((g / (2.0 * a)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cbrt((((a * a) * 0.5) * g)) / a;
	} else {
		tmp = Math.cbrt((g / (-1.0 / (-0.5 / a))));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0

   1. Initial program 4.4%

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

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

      5. cbrt-div N/A

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

      6. pow1/3 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. pow1/3 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. lower-/.f64 4.4

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 98.5%

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

   8. Applied rewrites 50.7%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 85.8%

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

      1. remove-double-div N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg2 N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. sqr-neg N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. pow-prod-down N/A

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

      17. pow-sqr N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied rewrites 85.8%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\frac{-1}{\frac{-0.5}{a}}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* 2.0 a))))
   (if (<= t_0 (- INFINITY)) (/ (cbrt (* (* (* a a) 0.5) g)) a) (cbrt t_0))))

C

double code(double g, double a) {
	double t_0 = g / (2.0 * a);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cbrt((((a * a) * 0.5) * g)) / a;
	} else {
		tmp = cbrt(t_0);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (2.0 * a);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cbrt((((a * a) * 0.5) * g)) / a;
	} else {
		tmp = Math.cbrt(t_0);
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cbrt(Float64(Float64(Float64(a * a) * 0.5) * g)) / a);
	else
		tmp = cbrt(t_0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0

   1. Initial program 4.4%

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

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

      5. cbrt-div N/A

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

      6. pow1/3 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. pow1/3 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. lower-/.f64 4.4

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 98.5%

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

   8. Applied rewrites 50.7%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 85.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.9% 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 80.7%

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

Alternative 8: 75.9% 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 80.7%

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

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

   4. metadata-eval N/A

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

   5. associate-/r/ N/A

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

   6. unpow-1 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. cbrt-div N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-cbrt.f64 79.9

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

   14. lift-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. unpow-1 N/A

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

   19. un-div-inv N/A

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

   20. associate-/r/ N/A

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

   21. lower-*.f64 N/A

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

6. Applied rewrites 80.7%

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

Reproduce

herbie shell --seed 2024323 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))