2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 5.5s

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: 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 92.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-306) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((g * -0.5));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	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((g * -0.5));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	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(g * -0.5)));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	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[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.3%

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

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 92.1

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

   6. Applied rewrites 92.1%

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

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

   1. Initial program 80.6%

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

   4. Applied rewrites 99.0%

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

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

   6. Applied rewrites 98.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.2

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

   8. Applied rewrites 92.2%

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

4. Final simplification 92.1%

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

Alternative 3: 84.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.3%

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

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

      4. lift-/.f64 N/A

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

      5. cbrt-div N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. cbrt-undiv N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. lift-neg.f64 N/A

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

      19. /-rgt-identity N/A

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

      20. lower-cbrt.f64 98.7

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-cbrt.f64 N/A

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

      14. div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. times-frac N/A

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

      19. neg-mul-1 N/A

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

      20. lift-neg.f64 N/A

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

      21. associate-/l/ N/A

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

      22. lift-neg.f64 N/A

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

      23. metadata-eval N/A

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

      24. frac-2neg N/A

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

      25. lower-/.f64 N/A

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

   8. Applied rewrites 81.6%

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

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

   1. Initial program 80.6%

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

   4. Applied rewrites 99.0%

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

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

   6. Applied rewrites 98.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. cbrt-unprod N/A

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

   12. neg-mul-1 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lift-neg.f64 N/A

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

   16. lower-cbrt.f64 N/A

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

   17. lift-neg.f64 N/A

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

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

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

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

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

   20. lower-*.f64 N/A

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

   21. metadata-eval 98.8

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

6. Applied rewrites 98.8%

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

Alternative 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 6: 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]

Derivation

1. Initial program 81.0%

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

Alternative 7: 76.3% 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 81.0%

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

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

   6. associate-*r* N/A

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

   7. cbrt-prod N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. mul-1-neg N/A

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

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

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

   12. associate-/r/ N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

   15. cbrt-prod N/A

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

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

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

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

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

   18. lift-neg.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-cbrt.f64 80.9

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

   21. lift-*.f64 N/A

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

   22. lift-/.f64 N/A

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

   23. associate-*l/ N/A

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

   24. frac-2neg N/A

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

6. Applied rewrites 80.9%

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

Reproduce

herbie shell --seed 2024283 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))