2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 9.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

   4. pow1/3 N/A

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

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

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

   6. pow1/3 N/A

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

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

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

   8. distribute-neg-frac2 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{pow.f64}\left(\left(0 - a\right), \frac{1}{3}\right)\right) \]

   14. --lowering--.f64 43.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, a\right), \frac{1}{3}\right)\right) \]

4. Applied egg-rr 43.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{-2}}}{{\left(0 - a\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{0 - a}\right)\right) \]

   2. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{\frac{{0}^{3} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{\frac{1}{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}\right)\right) \]

   4. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\frac{1}{\sqrt[3]{\color{blue}{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}\right)\right)\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{{0}^{3} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}\right)\right)\right)\right) \]

   9. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{0 - a}\right)\right)\right)\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(0 - a\right)\right)}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\left(0 - a\right)\right)}\right)\right)\right)\right) \]

   12. sub0-neg N/A

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

   13. remove-double-neg N/A

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

   14. /-lowering-/.f64 98.8%

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

6. Applied egg-rr 98.8%

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

   1. clear-num N/A

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

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

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

   3. /-lowering-/.f64 98.8%

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

8. Applied egg-rr 98.8%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

   4. pow1/3 N/A

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

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

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

   6. pow1/3 N/A

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

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

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

   8. distribute-neg-frac2 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{pow.f64}\left(\left(0 - a\right), \frac{1}{3}\right)\right) \]

   14. --lowering--.f64 43.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, a\right), \frac{1}{3}\right)\right) \]

4. Applied egg-rr 43.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{-2}}}{{\left(0 - a\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{0 - a}\right)\right) \]

   2. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{\frac{{0}^{3} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{\frac{1}{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}\right)\right) \]

   4. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\frac{1}{\sqrt[3]{\color{blue}{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}\right)\right)\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{{0}^{3} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}\right)\right)\right)\right) \]

   9. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{0 - a}\right)\right)\right)\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(0 - a\right)\right)}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\left(0 - a\right)\right)}\right)\right)\right)\right) \]

   12. sub0-neg N/A

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

   13. remove-double-neg N/A

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

   14. /-lowering-/.f64 98.8%

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

6. Applied egg-rr 98.8%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

   4. pow1/3 N/A

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

   5. neg-mul-1 N/A

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

   6. cbrt-prod N/A

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

   7. pow1/3 N/A

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

   8. associate-/r* N/A

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

   9. /-lowering-/.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. pow1/3 N/A

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

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

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

   13. distribute-neg-frac2 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. pow1/3 N/A

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

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

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

   18. cbrt-lowering-cbrt.f64 98.7%

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

4. Applied egg-rr 98.7%

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

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

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

   2. cbrt-undiv N/A

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

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

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

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

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

   7. cbrt-lowering-cbrt.f64 98.7%

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

6. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]
7. 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 79.4%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

   4. pow1/3 N/A

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

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

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

   6. pow1/3 N/A

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

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

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

   8. distribute-neg-frac2 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{pow.f64}\left(\left(0 - a\right), \frac{1}{3}\right)\right) \]

   14. --lowering--.f64 43.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, a\right), \frac{1}{3}\right)\right) \]

4. Applied egg-rr 43.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{-2}}}{{\left(0 - a\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{0 - a}\right)\right) \]

   2. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{\frac{{0}^{3} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\sqrt[3]{\frac{1}{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}\right)\right) \]

   4. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \left(\frac{1}{\sqrt[3]{\color{blue}{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}{{0}^{3} - {a}^{3}}\right)\right)\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{{0}^{3} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}\right)\right)\right)\right) \]

   9. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{0 - a}\right)\right)\right)\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(0 - a\right)\right)}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, -2\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\left(0 - a\right)\right)}\right)\right)\right)\right) \]

   12. sub0-neg N/A

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

   13. remove-double-neg N/A

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

   14. /-lowering-/.f64 98.8%

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

6. Applied egg-rr 98.8%

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

   1. clear-num N/A

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

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

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

   3. /-lowering-/.f64 98.8%

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

8. Applied egg-rr 98.8%

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

   1. associate-/r/ N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

   4. frac-times N/A

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

   5. clear-num N/A

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

   6. pow1/3 N/A

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

   7. cbrt-unprod N/A

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

   8. metadata-eval N/A

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

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

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

   10. *-rgt-identity N/A

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

   11. distribute-neg-frac2 N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. *-inverses N/A

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

   17. associate-/l* N/A

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

   18. *-commutative N/A

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

   19. associate-/l* N/A

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

   20. associate-/l* N/A

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

   21. associate-/l* N/A

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

   22. *-lft-identity N/A

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

10. Applied egg-rr 98.6%

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

Alternative 5: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-/r* N/A

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

   4. associate-/r/ N/A

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

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

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 79.2%

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

4. Applied egg-rr 79.2%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow2 N/A

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

   8. inv-pow N/A

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

   9. frac-times N/A

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

   10. associate-*l/ N/A

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

   11. associate-/r* N/A

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

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

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

   13. /-lowering-/.f64 N/A

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

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

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

   15. associate-/r/ N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 55.7%

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

6. Applied egg-rr 55.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

   9. /-lowering-/.f64 N/A

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

   10. /-lowering-/.f64 79.8%

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

8. Applied egg-rr 79.8%

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

Alternative 6: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. associate-/l/ N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-/l/ N/A

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

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

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

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

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

   7. /-lowering-/.f64 79.8%

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

4. Applied egg-rr 79.8%

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

Alternative 7: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 79.8%

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

3. Simplified 79.8%

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

Alternative 8: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. associate-/r* N/A

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

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

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

   6. metadata-eval 79.7%

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

4. Applied egg-rr 79.7%

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

5. Final simplification 79.7%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))