2-ancestry mixing, zero discriminant

Percentage Accurate: 76.2% → 98.6%

Time: 7.5s

Alternatives: 9

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: 76.2% 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

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

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 78.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

   12. neg-mul-1 N/A

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

   13. lift-neg.f64 N/A

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

   14. metadata-eval N/A

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

   15. associate-/r/ N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. cbrt-prod N/A

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

   19. lower-*.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. lift-/.f64 N/A

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

   22. associate-/r/ N/A

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

   23. metadata-eval N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 N/A

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

   26. lower-cbrt.f64 N/A

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

6. Applied rewrites 98.7%

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

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {\left(a \cdot -2\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -1e-304)
   (* (cbrt (- g)) (pow (* a -2.0) -0.3333333333333333))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -1e-304) {
		tmp = cbrt(-g) * pow((a * -2.0), -0.3333333333333333);
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -1e-304) {
		tmp = Math.cbrt(-g) * Math.pow((a * -2.0), -0.3333333333333333);
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -1e-304)
		tmp = Float64(cbrt(Float64(-g)) * (Float64(a * -2.0) ^ -0.3333333333333333));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -1e-304], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -9.99999999999999971e-305

   1. Initial program 76.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. times-frac N/A

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

      12. neg-mul-1 N/A

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

      13. lift-neg.f64 N/A

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

      14. metadata-eval N/A

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

      15. associate-/r/ N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. cbrt-prod N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cbrt.f64 N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. metadata-eval N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 N/A

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

      26. lower-cbrt.f64 N/A

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

   6. Applied rewrites 98.7%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. lift-cbrt.f64 N/A

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

      5. pow1/3 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. lower-cbrt.f64 98.7

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

   8. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-cbrt.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. cbrt-unprod N/A

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

      14. metadata-eval N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. associate-/r/ N/A

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

      18. clear-num N/A

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

      19. cbrt-div N/A

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

      20. associate-*l/ N/A

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

      21. metadata-eval N/A

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

      22. frac-2neg N/A

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

      23. lift-/.f64 N/A

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

      24. frac-2neg N/A

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

   10. Applied rewrites 92.2%

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

   if -9.99999999999999971e-305 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 91.6

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

   4. Applied rewrites 91.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {\left(a \cdot -2\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))