2-ancestry mixing, zero discriminant

Percentage Accurate: 76.1% → 98.7%

Time: 9.2s

Alternatives: 7

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: 76.1% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

   4. /-lowering-/.f64 74.5%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

3. Simplified 74.5%

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

   1. cbrt-div N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. frac-2neg N/A

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

   5. cbrt-div N/A

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

   6. associate-/r/ N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. cbrt-div N/A

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

   11. pow1/3 N/A

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

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}\right)}^{-1}\right), \color{blue}{\left({\left(\frac{1}{\mathsf{neg}\left(a\right)}\right)}^{\frac{1}{3}}\right)}\right) \]

6. Applied egg-rr 98.8%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

   4. /-lowering-/.f64 74.5%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

3. Simplified 74.5%

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

   1. cbrt-div N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. frac-2neg N/A

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

   5. cbrt-div N/A

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

   6. associate-/r/ N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. cbrt-div N/A

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

   11. pow1/3 N/A

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

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}\right)}^{-1}\right), \color{blue}{\left({\left(\frac{1}{\mathsf{neg}\left(a\right)}\right)}^{\frac{1}{3}}\right)}\right) \]

6. Applied egg-rr 98.8%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{-2}{g}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(-1, a\right)}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{-2}{g}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(-1, a\right)}\right)\right) \]

   3. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{-2}{g}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \color{blue}{a}\right)\right)\right) \]

   4. /-lowering-/.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-2, g\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, a\right)\right)\right) \]

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

   4. /-lowering-/.f64 74.5%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

3. Simplified 74.5%

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

   1. cbrt-div N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. frac-2neg N/A

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

   5. cbrt-div N/A

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

   6. associate-/r/ N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. cbrt-div N/A

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

   11. pow1/3 N/A

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

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}\right)}^{-1}\right), \color{blue}{\left({\left(\frac{1}{\mathsf{neg}\left(a\right)}\right)}^{\frac{1}{3}}\right)}\right) \]

6. Applied egg-rr 98.8%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{-2}{g}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(-1, a\right)}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{-2}{g}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(-1, a\right)}\right)\right) \]

   3. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{-2}{g}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \color{blue}{a}\right)\right)\right) \]

   4. /-lowering-/.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-2, g\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, a\right)\right)\right) \]

8. Applied egg-rr 98.8%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. cbrt-div N/A

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

   10. associate-/r/ N/A

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

   11. associate-*r/ N/A

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

   12. cbrt-div N/A

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

   13. frac-2neg N/A

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

   14. cbrt-div N/A

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

   15. frac-times N/A

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

   16. *-lft-identity N/A

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

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{a}\right)}\right) \]

   18. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\color{blue}{\sqrt[3]{2}} \cdot \sqrt[3]{a}\right)\right) \]

   19. cbrt-unprod N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\sqrt[3]{2 \cdot a}\right)\right) \]

   20. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(2 \cdot a\right)\right)\right) \]

   21. *-lowering-*.f64 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(2, a\right)\right)\right) \]

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

   4. /-lowering-/.f64 74.5%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

3. Simplified 74.5%

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

   1. cbrt-div N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. frac-2neg N/A

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

   5. cbrt-div N/A

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

   6. associate-/r/ N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. cbrt-div N/A

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

   11. pow1/3 N/A

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

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}\right)}^{-1}\right), \color{blue}{\left({\left(\frac{1}{\mathsf{neg}\left(a\right)}\right)}^{\frac{1}{3}}\right)}\right) \]

6. Applied egg-rr 98.8%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{-2}{g}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(-1, a\right)}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{-2}{g}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(-1, a\right)}\right)\right) \]

   3. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{-2}{g}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \color{blue}{a}\right)\right)\right) \]

   4. /-lowering-/.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-2, g\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, a\right)\right)\right) \]

8. Applied egg-rr 98.8%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. cbrt-div N/A

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

   10. associate-/r/ N/A

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

   11. associate-*r/ N/A

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

   12. cbrt-div N/A

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

   13. frac-2neg N/A

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

   14. cbrt-div N/A

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

   15. frac-times N/A

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

   16. *-lft-identity N/A

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

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{a}\right)}\right) \]

   18. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\color{blue}{\sqrt[3]{2}} \cdot \sqrt[3]{a}\right)\right) \]

   19. cbrt-unprod N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\sqrt[3]{2 \cdot a}\right)\right) \]

   20. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(2 \cdot a\right)\right)\right) \]

   21. *-lowering-*.f64 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(2, a\right)\right)\right) \]

10. Applied egg-rr 98.8%

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

    1. cbrt-undiv N/A

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

    2. div-inv N/A

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

    3. cbrt-prod N/A

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

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{\frac{1}{2 \cdot a}}\right)}\right) \]

    5. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}}}\right)\right) \]

    6. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{1}{2 \cdot a}\right)\right)\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{2}}{a}\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right)\right)\right) \]

    9. metadata-eval 98.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right) \]

12. Applied egg-rr 98.7%

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

Alternative 5: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{g \cdot \frac{0.5}{a}}\\ t_1 := \frac{g}{a \cdot 2}\\ t_2 := \frac{\sqrt[3]{\frac{a \cdot \left(g \cdot a\right)}{2}}}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-320}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (* g (/ 0.5 a))))
        (t_1 (/ g (* a 2.0)))
        (t_2 (/ (cbrt (/ (* a (* g a)) 2.0)) a)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-309)
       t_0
       (if (<= t_1 2e-320) t_2 (if (<= t_1 5e+293) t_0 t_2))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g * (0.5 / a)));
	double t_1 = g / (a * 2.0);
	double t_2 = cbrt(((a * (g * a)) / 2.0)) / a;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-309) {
		tmp = t_0;
	} else if (t_1 <= 2e-320) {
		tmp = t_2;
	} else if (t_1 <= 5e+293) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g * (0.5 / a)));
	double t_1 = g / (a * 2.0);
	double t_2 = Math.cbrt(((a * (g * a)) / 2.0)) / a;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= -2e-309) {
		tmp = t_0;
	} else if (t_1 <= 2e-320) {
		tmp = t_2;
	} else if (t_1 <= 5e+293) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g * Float64(0.5 / a)))
	t_1 = Float64(g / Float64(a * 2.0))
	t_2 = Float64(cbrt(Float64(Float64(a * Float64(g * a)) / 2.0)) / a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-309)
		tmp = t_0;
	elseif (t_1 <= 2e-320)
		tmp = t_2;
	elseif (t_1 <= 5e+293)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(a * N[(g * a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-309], t$95$0, If[LessEqual[t$95$1, 2e-320], t$95$2, If[LessEqual[t$95$1, 5e+293], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -1.9999999999999988e-309 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.99998e-320 or 5.00000000000000033e293 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 5.0%

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

      1. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

      4. /-lowering-/.f64 5.0%

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

   3. Simplified 5.0%

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

      1. cbrt-div N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. frac-2neg N/A

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

      5. cbrt-div N/A

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

      6. associate-/r/ N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. cbrt-div N/A

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

      11. pow1/3 N/A

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

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}\right)}^{-1}\right), \color{blue}{\left({\left(\frac{1}{\mathsf{neg}\left(a\right)}\right)}^{\frac{1}{3}}\right)}\right) \]

   6. Applied egg-rr 98.8%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. cbrt-div N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. sub0-neg N/A

         \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{0 - a}} \]

      8. flip-- N/A

         \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0 \cdot 0 - a \cdot a}{0 + a}}} \]

      9. +-lft-identity N/A

         \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0 \cdot 0 - a \cdot a}{a}}} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0 - a \cdot a}{a}}} \]

      11. sub-div N/A

          \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0}{a} - \frac{a \cdot a}{a}}} \]

      12. frac-sub N/A

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

      13. mul0-lft N/A

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

      14. clear-num N/A

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

      15. cbrt-prod N/A

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

      16. associate-/l* N/A

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

   8. Applied egg-rr 24.3%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\left(g \cdot a\right) \cdot a\right), 2\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\left(a \cdot g\right) \cdot a\right), 2\right)\right), a\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot g\right), a\right), 2\right)\right), a\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(g \cdot a\right), a\right), 2\right)\right), a\right) \]

      5. *-lowering-*.f64 67.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(g, a\right), a\right), 2\right)\right), a\right) \]

   10. Applied egg-rr 67.4%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999988e-309 or 1.99998e-320 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 5.00000000000000033e293

   1. Initial program 99.1%

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

      1. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

      4. /-lowering-/.f64 99.1%

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

   3. Simplified 99.1%

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

      1. div-inv N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g \cdot \frac{1}{a}}{2}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{\frac{1}{a}}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{a}}{2} \cdot g\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{a}}{2}\right), g\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

      8. metadata-eval 99.2%

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

   6. Applied egg-rr 99.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a \cdot \left(g \cdot a\right)}{2}}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a \cdot \left(g \cdot a\right)}{2}}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a \cdot \left(g \cdot a\right)}{2}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

      1. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

      4. /-lowering-/.f64 4.4%

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

   3. Simplified 4.4%

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

      1. cbrt-div N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. frac-2neg N/A

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

      5. cbrt-div N/A

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

      6. associate-/r/ N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. cbrt-div N/A

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

      11. pow1/3 N/A

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

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{neg}\left(g\right)}}\right)}^{-1}\right), \color{blue}{\left({\left(\frac{1}{\mathsf{neg}\left(a\right)}\right)}^{\frac{1}{3}}\right)}\right) \]

   6. Applied egg-rr 98.9%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. cbrt-div N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. sub0-neg N/A

         \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{0 - a}} \]

      8. flip-- N/A

         \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0 \cdot 0 - a \cdot a}{0 + a}}} \]

      9. +-lft-identity N/A

         \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0 \cdot 0 - a \cdot a}{a}}} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0 - a \cdot a}{a}}} \]

      11. sub-div N/A

          \[\leadsto \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{1}{\frac{0}{a} - \frac{a \cdot a}{a}}} \]

      12. frac-sub N/A

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

      13. mul0-lft N/A

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

      14. clear-num N/A

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

      15. cbrt-prod N/A

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

      16. associate-/l* N/A

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

   8. Applied egg-rr 27.7%

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

      1. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g \cdot \left(a \cdot a\right)}{2}\right)\right), a\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(g \cdot \frac{a \cdot a}{2}\right)\right), a\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{\frac{2}{a \cdot a}}\right)\right), a\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{\frac{2}{a \cdot a}}\right)\right), a\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, \left(\frac{2}{a \cdot a}\right)\right)\right), a\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, \mathsf{/.f64}\left(2, \left(a \cdot a\right)\right)\right)\right), a\right) \]

      7. *-lowering-*.f64 27.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(a, a\right)\right)\right)\right), a\right) \]

   10. Applied egg-rr 27.8%

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

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

   1. Initial program 80.1%

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

      1. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

      4. /-lowering-/.f64 80.1%

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

   3. Simplified 80.1%

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

      1. div-inv N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g \cdot \frac{1}{a}}{2}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{\frac{1}{a}}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{a}}{2} \cdot g\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{a}}{2}\right), g\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

      8. metadata-eval 80.2%

         \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

   6. Applied egg-rr 80.2%

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

4. Final simplification 76.3%

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

Alternative 7: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{2 \cdot a}\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{a}}{2}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{g}{a}\right), 2\right)\right) \]

   4. /-lowering-/.f64 74.5%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(g, a\right), 2\right)\right) \]

3. Simplified 74.5%

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

   1. div-inv N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g \cdot \frac{1}{a}}{2}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{\frac{1}{a}}{2}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{a}}{2} \cdot g\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{a}}{2}\right), g\right)\right) \]

   5. associate-/l/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

   8. metadata-eval 74.5%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

6. Applied egg-rr 74.5%

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

7. Final simplification 74.5%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))