2-ancestry mixing, zero discriminant

Percentage Accurate: 75.6% → 98.7%

Time: 2.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 75.6% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sqrt[3]{g}}{{2}^{0.3333333333333333}}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (/ (/ (cbrt g) (pow 2.0 0.3333333333333333)) (cbrt a)))

C

double code(double g, double a) {
	return (cbrt(g) / pow(2.0, 0.3333333333333333)) / cbrt(a);
}

Java

public static double code(double g, double a) {
	return (Math.cbrt(g) / Math.pow(2.0, 0.3333333333333333)) / Math.cbrt(a);
}

Julia

function code(g, a)
	return Float64(Float64(cbrt(g) / (2.0 ^ 0.3333333333333333)) / cbrt(a))
end

Wolfram

code[g_, a_] := N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.1

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

6. Applied rewrites 98.1%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 98.7

      \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{{2}^{0.3333333333333333}}}}{\sqrt[3]{a}} \]

8. Applied rewrites 98.7%

   \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{{2}^{0.3333333333333333}}}}{\sqrt[3]{a}} \]
9. Add Preprocessing

Alternative 2: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-304}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g}}{{\left(a + a\right)}^{0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= a 3.5e-304)
   (* (cbrt (/ g a)) (cbrt 0.5))
   (/ (cbrt g) (pow (+ a a) 0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if (a <= 3.5e-304) {
		tmp = cbrt((g / a)) * cbrt(0.5);
	} else {
		tmp = cbrt(g) / pow((a + a), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (a <= 3.5e-304) {
		tmp = Math.cbrt((g / a)) * Math.cbrt(0.5);
	} else {
		tmp = Math.cbrt(g) / Math.pow((a + a), 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (a <= 3.5e-304)
		tmp = Float64(cbrt(Float64(g / a)) * cbrt(0.5));
	else
		tmp = Float64(cbrt(g) / (Float64(a + a) ^ 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[a, 3.5e-304], N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[N[(a + a), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.5e-304

   1. Initial program 79.8%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lower-cbrt.f64 98.1

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

   6. Applied rewrites 98.1%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 98.8

         \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{{2}^{0.3333333333333333}}}}{\sqrt[3]{a}} \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{{2}^{0.3333333333333333}}}}{\sqrt[3]{a}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/3 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. frac-times N/A

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

      11. cbrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. cbrt-unprod N/A

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

      15. associate-/l* N/A

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

      16. cbrt-undiv N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. cbrt-unprod N/A

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

      20. associate-*l* N/A

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

      21. cbrt-unprod N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   10. Applied rewrites 80.6%

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

   if 3.5e-304 < a

   1. Initial program 79.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lower-pow.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 92.1

         \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(a \cdot 2\right)}}^{0.3333333333333333}} \]

   6. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 92.1

         \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(a + a\right)}}^{0.3333333333333333}} \]

   8. Applied rewrites 92.1%

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(a + a\right)}}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-304}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot \frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g}}{{\left(a + a\right)}^{0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= a 3.5e-304)
   (cbrt (* 0.5 (/ g a)))
   (/ (cbrt g) (pow (+ a a) 0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if (a <= 3.5e-304) {
		tmp = cbrt((0.5 * (g / a)));
	} else {
		tmp = cbrt(g) / pow((a + a), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (a <= 3.5e-304) {
		tmp = Math.cbrt((0.5 * (g / a)));
	} else {
		tmp = Math.cbrt(g) / Math.pow((a + a), 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (a <= 3.5e-304)
		tmp = cbrt(Float64(0.5 * Float64(g / a)));
	else
		tmp = Float64(cbrt(g) / (Float64(a + a) ^ 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[a, 3.5e-304], N[Power[N[(0.5 * N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[N[(a + a), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.5e-304

   1. Initial program 79.8%

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

   3. Taylor expanded in g around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if 3.5e-304 < a

   1. Initial program 79.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lower-pow.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 92.1

         \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(a \cdot 2\right)}}^{0.3333333333333333}} \]

   6. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 92.1

         \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(a + a\right)}}^{0.3333333333333333}} \]

   8. Applied rewrites 92.1%

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(a + a\right)}}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in g around 0

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

   1. lower-*.f64 98.7

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

7. Applied rewrites 98.7%

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

Alternative 5: 98.6% 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 79.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. cbrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 6: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in g around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 79.7

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

5. Applied rewrites 79.7%

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

Alternative 7: 75.6% 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 79.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. 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 79.3

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

4. Applied rewrites 79.3%

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

Alternative 8: 4.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.1

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

6. Applied rewrites 98.1%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 98.7

      \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{{2}^{0.3333333333333333}}}}{\sqrt[3]{a}} \]

8. Applied rewrites 98.7%

   \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{{2}^{0.3333333333333333}}}}{\sqrt[3]{a}} \]

9. Taylor expanded in g around -inf

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

    1. pow1/3 N/A

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

    2. cbrt-div N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. cbrt-undiv N/A

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

    6. metadata-eval N/A

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

    7. cbrt-unprod N/A

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

    8. metadata-eval N/A

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

    9. frac-times N/A

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

    10. *-commutative N/A

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

    11. count-2-rev N/A

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

    12. flip-+ N/A

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

    13. +-inverses N/A

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

    14. metadata-eval N/A

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

    15. metadata-eval N/A

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

    16. metadata-eval N/A

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

    17. +-inverses N/A

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

    18. metadata-eval N/A

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

    19. flip-+ N/A

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

    20. metadata-eval N/A

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

    21. cbrt-undiv N/A

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

11. Applied rewrites 5.2%

    \[\leadsto \color{blue}{-\sqrt[3]{-0.5 \cdot g}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025054 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))