2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 7.0s

Alternatives: 6

Speedup: 0.5×

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

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{\frac{4}{\frac{2}{a}}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt g) (cbrt (/ 4.0 (/ 2.0 a)))))

C

double code(double g, double a) {
	return cbrt(g) / cbrt((4.0 / (2.0 / a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(g) / Math.cbrt((4.0 / (2.0 / a)));
}

Julia

function code(g, a)
	return Float64(cbrt(g) / cbrt(Float64(4.0 / Float64(2.0 / a))))
end

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[N[(4.0 / N[(2.0 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

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

   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 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-/r/ N/A

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

   4. div-inv N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 2: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -4 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -4e-307)
   (* (pow (- a) -0.3333333333333333) (cbrt (* g -0.5)))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -4e-307) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((g * -0.5));
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -4e-307) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((g * -0.5));
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -4e-307)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(g * -0.5)));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -4e-307], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -3.99999999999999964e-307

   1. Initial program 79.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. cbrt-prod N/A

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

      6. *-rgt-identity N/A

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

      7. *-inverses N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. cbrt-prod N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

          \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{\frac{4 \cdot \left(a \cdot a\right)}{2 \cdot a}}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{2 \cdot a}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\frac{\color{blue}{\left(a \cdot a\right) \cdot 4}}{2 \cdot a}}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 4}{\color{blue}{2 \cdot a}}}} \]

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. cbrt-div N/A

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

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\sqrt[3]{g \cdot 2} \cdot \sqrt[3]{\frac{0.25}{a}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. associate-*l/ N/A

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

      13. frac-2neg N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(a\right)}} \cdot \frac{g}{4}} \]

      14. associate-*l/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{g}{4}}{\mathsf{neg}\left(a\right)}}} \]

      15. clear-num N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(a\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{g}{4}}}}} \]

      16. associate-/r/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\mathsf{neg}\left(a\right)} \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{g}{4}\right)}} \]

      17. cbrt-prod N/A

          \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\mathsf{neg}\left(a\right)}} \cdot \sqrt[3]{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{g}{4}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\mathsf{neg}\left(a\right)}} \cdot \sqrt[3]{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{g}{4}}} \]

   8. Applied rewrites 92.4%

      \[\leadsto \color{blue}{{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}} \]

   if -3.99999999999999964e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 72.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.2

          \[\leadsto {a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot \color{blue}{0.5}} \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -4 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -4e-307)
   (cbrt (/ (/ 1.0 a) (/ 2.0 g)))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -4e-307) {
		tmp = cbrt(((1.0 / a) / (2.0 / g)));
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -4e-307) {
		tmp = Math.cbrt(((1.0 / a) / (2.0 / g)));
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -4e-307)
		tmp = cbrt(Float64(Float64(1.0 / a) / Float64(2.0 / g)));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -4e-307], N[Power[N[(N[(1.0 / a), $MachinePrecision] / N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -3.99999999999999964e-307

   1. Initial program 79.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

   if -3.99999999999999964e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 72.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.2

          \[\leadsto {a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot \color{blue}{0.5}} \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

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

   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 98.8

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

4. Applied rewrites 98.8%

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

Alternative 5: 76.2% accurate, 0.9× 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 76.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 76.0

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

4. Applied rewrites 76.0%

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

Alternative 6: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 76.0

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

4. Applied rewrites 76.0%

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

5. Final simplification 76.0%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))