2-ancestry mixing, zero discriminant

Percentage Accurate: 76.7% → 98.7%

Time: 5.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.7% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. pow1/3 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. pow1/3 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. pow1/3 N/A

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

   14. lower-cbrt.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-306}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2 \cdot a}{g}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(2 \cdot a\right)}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -1e-306)
   (pow (cbrt (/ (* 2.0 a) g)) -1.0)
   (* (cbrt g) (pow (* 2.0 a) -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-306) {
		tmp = pow(cbrt(((2.0 * a) / g)), -1.0);
	} else {
		tmp = cbrt(g) * pow((2.0 * a), -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-306) {
		tmp = Math.pow(Math.cbrt(((2.0 * a) / g)), -1.0);
	} else {
		tmp = Math.cbrt(g) * Math.pow((2.0 * a), -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -1e-306)
		tmp = cbrt(Float64(Float64(2.0 * a) / g)) ^ -1.0;
	else
		tmp = Float64(cbrt(g) * (Float64(2.0 * a) ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -1e-306], N[Power[N[Power[N[(N[(2.0 * a), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision], -1.0], $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.00000000000000003e-306

   1. Initial program 78.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. *-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. pow1/3 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/3 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow1/3 N/A

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

      14. lower-cbrt.f64 98.7

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

   6. Applied rewrites 98.7%

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

   8. Applied rewrites 79.8%

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

   if -1.00000000000000003e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. pow1/3 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/3 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow1/3 N/A

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

      14. lower-cbrt.f64 98.9

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

   6. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{\frac{-1}{a}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. inv-pow N/A

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

      11. unpow-prod-down N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. pow-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 92.2

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

   8. Applied rewrites 92.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-306}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2 \cdot a}{g}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(2 \cdot a\right)}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-306}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(2 \cdot a\right)}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -1e-306)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (* (cbrt g) (pow (* 2.0 a) -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-306) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = cbrt(g) * pow((2.0 * a), -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-306) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = Math.cbrt(g) * Math.pow((2.0 * a), -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -1e-306)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = Float64(cbrt(g) * (Float64(2.0 * a) ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -1e-306], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.00000000000000003e-306

   1. Initial program 78.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

         \[\leadsto {\left(\mathsf{neg}\left(a\right)\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{6}\right)}} \cdot \sqrt[3]{\frac{-1}{2} \cdot g} \]

      10. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(a\right)\right)}^{\left(2 \cdot \frac{-1}{6}\right)}} \cdot \sqrt[3]{\frac{-1}{2} \cdot g} \]

      11. lower-neg.f64 N/A

          \[\leadsto {\color{blue}{\left(-a\right)}}^{\left(2 \cdot \frac{-1}{6}\right)} \cdot \sqrt[3]{\frac{-1}{2} \cdot g} \]

      12. 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 -1.00000000000000003e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. pow1/3 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/3 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow1/3 N/A

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

      14. lower-cbrt.f64 98.9

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

   6. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{\frac{-1}{a}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. inv-pow N/A

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

      11. unpow-prod-down N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. pow-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

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 77.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 5: 76.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. pow1/3 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. pow1/3 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. pow1/3 N/A

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

   14. lower-cbrt.f64 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 78.3%

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

9. Final simplification 78.3%

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

Alternative 6: 76.5% 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 77.4%

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 77.9%

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

7. Final simplification 77.9%

   \[\leadsto {\left(\sqrt[3]{\frac{2}{g} \cdot a}\right)}^{-1} \]
8. Add Preprocessing

Alternative 7: 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 77.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. pow1/3 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow-prod-down N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. pow1/3 N/A

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

   10. cbrt-unprod N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l/ N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. distribute-neg-frac N/A

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

   16. lift-/.f64 N/A

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

   17. lower-cbrt.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. distribute-neg-frac N/A

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

   20. lower-/.f64 N/A

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

   21. metadata-eval N/A

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

   22. pow1/3 N/A

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

   23. lower-cbrt.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 8: 76.7% 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 77.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. cbrt-div N/A

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

   10. *-lft-identity N/A

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

   11. remove-double-neg N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lift-neg.f64 N/A

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

   15. cbrt-div N/A

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

   16. frac-2neg N/A

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

   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-*.f64 N/A

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

   20. lift-cbrt.f64 77.4

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

   21. lift-*.f64 N/A

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

6. Applied rewrites 77.4%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))