2-ancestry mixing, zero discriminant

Percentage Accurate: 76.8% → 98.6%

Time: 6.2s

Alternatives: 4

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: 76.8% 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: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

4. Applied rewrites 98.6%

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

6. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 84.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -2e-306)
   (* (cbrt (* -0.5 g)) (pow (- a) -0.3333333333333333))
   (cbrt (* g (/ 0.5 a)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-306) {
		tmp = cbrt((-0.5 * g)) * pow(-a, -0.3333333333333333);
	} else {
		tmp = cbrt((g * (0.5 / a)));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-306) {
		tmp = Math.cbrt((-0.5 * g)) * Math.pow(-a, -0.3333333333333333);
	} else {
		tmp = Math.cbrt((g * (0.5 / a)));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -2e-306)
		tmp = Float64(cbrt(Float64(-0.5 * g)) * (Float64(-a) ^ -0.3333333333333333));
	else
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -2e-306], N[(N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -2.00000000000000006e-306

   1. Initial program 78.1%

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

   4. Applied rewrites 98.6%

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

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.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 -2.00000000000000006e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 72.4%

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

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

      7. neg-mul-1 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cbrt.f64 72.4

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. frac-2neg N/A

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

      18. metadata-eval N/A

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

      19. associate-/r/ N/A

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

      20. lift-neg.f64 N/A

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

      21. frac-2neg N/A

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

      22. associate-/r/ N/A

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

      23. lower-*.f64 N/A

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

      24. lower-/.f64 72.4

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

   6. Applied rewrites 72.4%

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

4. Final simplification 81.6%

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

Alternative 3: 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 75.0%

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
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. associate-*l/ N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-*r* N/A

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

   14. neg-mul-1 N/A

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

   15. lift-neg.f64 N/A

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

   16. cbrt-div N/A

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

   17. associate-*l/ N/A

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

   18. associate-/r/ N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. associate-/r/ N/A

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

   22. cbrt-prod N/A

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

   23. lower-*.f64 N/A

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

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 4: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
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. associate-*l/ N/A

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

   7. neg-mul-1 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lift-neg.f64 N/A

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

   11. associate-*l/ N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-cbrt.f64 75.0

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

   15. lift-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. frac-2neg N/A

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

   18. metadata-eval N/A

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

   19. associate-/r/ N/A

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

   20. lift-neg.f64 N/A

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

   21. frac-2neg N/A

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

   22. associate-/r/ N/A

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

   23. lower-*.f64 N/A

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

   24. lower-/.f64 75.0

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

6. Applied rewrites 75.0%

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

7. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))