2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 5.0s

Alternatives: 7

Speedup: 0.6×

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

\[\begin{array}{l} \\ \frac{1}{\frac{\sqrt[3]{4 \cdot a}}{\sqrt[3]{g + g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ 1.0 (/ (cbrt (* 4.0 a)) (cbrt (+ g g)))))

C

double code(double g, double a) {
	return 1.0 / (cbrt((4.0 * a)) / cbrt((g + g)));
}

Java

public static double code(double g, double a) {
	return 1.0 / (Math.cbrt((4.0 * a)) / Math.cbrt((g + g)));
}

Julia

function code(g, a)
	return Float64(1.0 / Float64(cbrt(Float64(4.0 * a)) / cbrt(Float64(g + g))))
end

Wolfram

code[g_, a_] := N[(1.0 / N[(N[Power[N[(4.0 * a), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. lift-cbrt.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. mult-flip N/A

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

   9. frac-times N/A

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

   10. cbrt-div N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. *-commutative N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 N/A

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

   17. lower-cbrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 98.6

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

   20. lift-*.f64 N/A

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

   21. count-2-rev N/A

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

   22. lower-+.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-special-/.f64 N/A

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

   4. lower-special-/.f64 98.6

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. count-2 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 98.6

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

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{4 \cdot a}}{\sqrt[3]{g + g}}}} \]
6. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g + g}}{\sqrt[3]{a \cdot 4}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (+ g g)) (cbrt (* a 4.0))))

C

double code(double g, double a) {
	return cbrt((g + g)) / cbrt((a * 4.0));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g + g)) / Math.cbrt((a * 4.0));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(g + g)) / cbrt(Float64(a * 4.0)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * 4.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. lift-cbrt.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. mult-flip N/A

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

   9. frac-times N/A

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

   10. cbrt-div N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. *-commutative N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 N/A

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

   17. lower-cbrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 98.6

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

   20. lift-*.f64 N/A

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

   21. count-2-rev N/A

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

   22. lower-+.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. count-2 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 98.6

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

5. Applied rewrites 98.6%

   \[\leadsto \frac{\sqrt[3]{g + g}}{\sqrt[3]{\color{blue}{a \cdot 4}}} \]
6. Add Preprocessing

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g + g} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (* (cbrt (/ 0.25 a)) (cbrt (+ g g))))

C

double code(double g, double a) {
	return cbrt((0.25 / a)) * cbrt((g + g));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((0.25 / a)) * Math.cbrt((g + g));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(0.25 / a)) * cbrt(Float64(g + g)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(0.25 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. lift-cbrt.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. mult-flip N/A

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

   9. frac-times N/A

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

   10. cbrt-div N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. *-commutative N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 N/A

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

   17. lower-cbrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 98.6

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

   20. lift-*.f64 N/A

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

   21. count-2-rev N/A

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

   22. lower-+.f64 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. inv-pow N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow-cbrt N/A

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

   8. cbrt-pow N/A

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

   9. inv-pow N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. count-2 N/A

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

   15. associate-*r* N/A

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

   16. associate-/r* N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f64 N/A

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

   21. metadata-eval 98.7

       \[\leadsto \sqrt[3]{\frac{\color{blue}{0.25}}{a}} \cdot \sqrt[3]{g + g} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g + g}} \]
6. Add Preprocessing

Alternative 4: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

   7. lift-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 98.7

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

3. Applied rewrites 98.7%

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

Alternative 5: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ \mathbf{if}\;t\_0 \leq 10^{-103}:\\ \;\;\;\;e^{\left(\log \left(0.5 \cdot g\right) - \log a\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 10^{+102}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))))
   (if (<= t_0 1e-103)
     (exp (* (- (log (* 0.5 g)) (log a)) 0.3333333333333333))
     (if (<= t_0 1e+102)
       (/ 1.0 (cbrt (/ (+ a a) g)))
       (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 1e-103) {
		tmp = exp(((log((0.5 * g)) - log(a)) * 0.3333333333333333));
	} else if (t_0 <= 1e+102) {
		tmp = 1.0 / cbrt(((a + a) / g));
	} else {
		tmp = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 1e-103) {
		tmp = Math.exp(((Math.log((0.5 * g)) - Math.log(a)) * 0.3333333333333333));
	} else if (t_0 <= 1e+102) {
		tmp = 1.0 / Math.cbrt(((a + a) / g));
	} else {
		tmp = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	tmp = 0.0
	if (t_0 <= 1e-103)
		tmp = exp(Float64(Float64(log(Float64(0.5 * g)) - log(a)) * 0.3333333333333333));
	elseif (t_0 <= 1e+102)
		tmp = Float64(1.0 / cbrt(Float64(Float64(a + a) / g)));
	else
		tmp = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$0, 1e-103], N[Exp[N[(N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+102], N[(1.0 / N[Power[N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.99999999999999958e-104

   1. Initial program 76.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-special-exp.f64 N/A

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

      5. lower-special-*.f64 N/A

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

      6. lower-special-log.f64 36.5

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.5

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 36.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-/r* N/A

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

      6. log-div N/A

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

      7. lower-special--.f64 N/A

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

      8. lower-special-log.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-special-log.f64 23.3

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

   5. Applied rewrites 23.3%

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

   if 9.99999999999999958e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.99999999999999977e101

   1. Initial program 76.3%

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

      1. lift-cbrt.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. mult-flip N/A

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

      9. frac-times N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 98.6

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

      20. lift-*.f64 N/A

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

      21. count-2-rev N/A

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

      22. lower-+.f64 98.6

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

   3. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-special-/.f64 N/A

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

      4. lower-special-/.f64 98.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{4 \cdot a}}{\sqrt[3]{g + g}}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

   if 9.99999999999999977e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-special-exp.f64 N/A

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

      5. lower-special-*.f64 N/A

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

      6. lower-special-log.f64 36.5

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.5

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 36.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower-special--.f64 N/A

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

      5. lower-special-log.f64 N/A

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

      6. lower-special-log.f64 23.3

         \[\leadsto e^{\left(\log g - \color{blue}{\log \left(a + a\right)}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 23.3%

      \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 43.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{if}\;t\_0 \leq 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+102}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a))))
        (t_1 (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))))
   (if (<= t_0 1e-103)
     t_1
     (if (<= t_0 1e+102) (/ 1.0 (cbrt (/ (+ a a) g))) t_1))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 1e-103) {
		tmp = t_1;
	} else if (t_0 <= 1e+102) {
		tmp = 1.0 / cbrt(((a + a) / g));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double t_1 = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 1e-103) {
		tmp = t_1;
	} else if (t_0 <= 1e+102) {
		tmp = 1.0 / Math.cbrt(((a + a) / g));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333))
	tmp = 0.0
	if (t_0 <= 1e-103)
		tmp = t_1;
	elseif (t_0 <= 1e+102)
		tmp = Float64(1.0 / cbrt(Float64(Float64(a + a) / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1e-103], t$95$1, If[LessEqual[t$95$0, 1e+102], N[(1.0 / N[Power[N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.99999999999999958e-104 or 9.99999999999999977e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-special-exp.f64 N/A

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

      5. lower-special-*.f64 N/A

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

      6. lower-special-log.f64 36.5

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 36.5

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 36.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower-special--.f64 N/A

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

      5. lower-special-log.f64 N/A

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

      6. lower-special-log.f64 23.3

         \[\leadsto e^{\left(\log g - \color{blue}{\log \left(a + a\right)}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 23.3%

      \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot 0.3333333333333333} \]

   if 9.99999999999999958e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.99999999999999977e101

   1. Initial program 76.3%

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

      1. lift-cbrt.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. mult-flip N/A

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

      9. frac-times N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 98.6

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

      20. lift-*.f64 N/A

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

      21. count-2-rev N/A

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

      22. lower-+.f64 98.6

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

   3. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-special-/.f64 N/A

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

      4. lower-special-/.f64 98.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{4 \cdot a}}{\sqrt[3]{g + g}}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

Alternative 7: 43.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 76.3

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

3. Applied rewrites 76.3%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))