2-ancestry mixing, zero discriminant

Percentage Accurate: 75.6% → 98.7%

Time: 5.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (* (cbrt (/ -1.0 a)) (cbrt (* -0.5 g))))

C

double code(double g, double a) {
	return cbrt((-1.0 / a)) * cbrt((-0.5 * g));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((-1.0 / a)) * Math.cbrt((-0.5 * g));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(-1.0 / a)) * cbrt(Float64(-0.5 * g)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-305}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{a}{g} \cdot 2}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-305)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (pow (cbrt (* (/ a g) 2.0)) -1.0)))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-305) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = pow(cbrt(((a / g) * 2.0)), -1.0);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-305) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = Math.pow(Math.cbrt(((a / g) * 2.0)), -1.0);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -5e-305)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = cbrt(Float64(Float64(a / g) * 2.0)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -5e-305], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(a / g), $MachinePrecision] * 2.0), $MachinePrecision], 1/3], $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.99999999999999985e-305

   1. Initial program 68.5%

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

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval 92.0

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

   6. Applied rewrites 92.0%

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

   if -4.99999999999999985e-305 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 71.6%

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

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-/r/ N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-cbrt.f64 N/A

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

      15. lift-cbrt.f64 N/A

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

      16. cbrt-undiv N/A

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

      17. lower-cbrt.f64 N/A

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

      18. lift-neg.f64 N/A

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

      19. neg-mul-1 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. times-frac N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 N/A

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

      26. lower-/.f64 72.6

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

   6. Applied rewrites 72.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-305}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{a}{g} \cdot 2}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. cbrt-prod N/A

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

   6. pow1/3 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. pow1/3 N/A

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

   12. lower-cbrt.f64 98.7

       \[\leadsto \left(\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{g}\right) \cdot \color{blue}{\sqrt[3]{-0.5}} \]

6. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{g}\right) \cdot \sqrt[3]{-0.5}} \]

8. Applied rewrites 70.3%

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

9. Final simplification 70.4%

   \[\leadsto {\left(\sqrt[3]{\frac{2}{g} \cdot a}\right)}^{-1} \]
10. 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 69.9%

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-unprod N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lift-neg.f64 N/A

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

   10. div-inv N/A

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

   11. lift-neg.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. distribute-neg-frac N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. lift-neg.f64 N/A

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

   17. cbrt-div N/A

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

   18. lower-/.f64 N/A

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

6. Applied rewrites 98.7%

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

Alternative 5: 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 69.9%

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. unpow-prod-down N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. times-frac N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. unpow-prod-down N/A

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

   22. neg-mul-1 N/A

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

   23. pow1/3 N/A

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

6. Applied rewrites 98.6%

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

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-unprod N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lift-neg.f64 N/A

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

   10. div-inv N/A

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

   11. lift-neg.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. distribute-neg-frac N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. lift-neg.f64 N/A

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

   17. associate-*l/ N/A

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

   18. lift-/.f64 N/A

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

   19. lift-neg.f64 N/A

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

   20. distribute-rgt-neg-out N/A

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

   21. distribute-lft-neg-in N/A

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

   22. cbrt-prod N/A

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

   23. lower-*.f64 N/A

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

6. Applied rewrites 98.6%

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

Alternative 7: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow2 N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

4. Applied rewrites 70.0%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. distribute-neg-frac N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 70.0

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

6. Applied rewrites 70.0%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))