2-ancestry mixing, zero discriminant

Percentage Accurate: 76.5% → 98.7%

Time: 2.3s

Alternatives: 5

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: 76.5% 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

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

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 77.6%

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

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in g around 0

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{2} \cdot g}}}{\sqrt[3]{a}} \]
6. Step-by-step derivation

   1. lower-*.f64 98.7

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

7. Applied rewrites 98.7%

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

Alternative 2: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{g}^{0.3333333333333333}}{\sqrt[3]{a + a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= g 6.5e-299)
   (cbrt (/ g (+ a a)))
   (/ (pow g 0.3333333333333333) (cbrt (+ a a)))))

C

double code(double g, double a) {
	double tmp;
	if (g <= 6.5e-299) {
		tmp = cbrt((g / (a + a)));
	} else {
		tmp = pow(g, 0.3333333333333333) / cbrt((a + a));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (g <= 6.5e-299) {
		tmp = Math.cbrt((g / (a + a)));
	} else {
		tmp = Math.pow(g, 0.3333333333333333) / Math.cbrt((a + a));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (g <= 6.5e-299)
		tmp = cbrt(Float64(g / Float64(a + a)));
	else
		tmp = Float64((g ^ 0.3333333333333333) / cbrt(Float64(a + a)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[g, 6.5e-299], N[Power[N[(g / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[g, 0.3333333333333333], $MachinePrecision] / N[Power[N[(a + a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < 6.4999999999999997e-299

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 74.2

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

   4. Applied rewrites 74.2%

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

   if 6.4999999999999997e-299 < g

   1. Initial program 81.2%

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

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

      3. lift-/.f64 N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 92.5

         \[\leadsto \frac{\color{blue}{{g}^{0.3333333333333333}}}{\sqrt[3]{a \cdot 2}} \]

   6. Applied rewrites 92.5%

      \[\leadsto \frac{\color{blue}{{g}^{0.3333333333333333}}}{\sqrt[3]{a \cdot 2}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 92.5

         \[\leadsto \frac{{g}^{0.3333333333333333}}{\sqrt[3]{\color{blue}{a + a}}} \]

   8. Applied rewrites 92.5%

      \[\leadsto \frac{{g}^{0.3333333333333333}}{\sqrt[3]{\color{blue}{a + a}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

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

   3. lift-/.f64 N/A

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

   4. cbrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.7

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

4. Applied rewrites 98.7%

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

Alternative 4: 76.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (+ a a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 77.6

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

4. Applied rewrites 77.6%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]
5. Add Preprocessing

Alternative 5: 4.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in g around 0

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{2} \cdot g}}}{\sqrt[3]{a}} \]
6. Step-by-step derivation

   1. lower-*.f64 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 45.0

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

9. Applied rewrites 45.0%

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

10. Taylor expanded in g around 0

    \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2}}} \]
11. Step-by-step derivation

    1. cbrt-unprod N/A

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

    2. metadata-eval N/A

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

    3. frac-times N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

    7. count-2-rev N/A

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

    8. flip-+ N/A

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

    9. +-inverses N/A

       \[\leadsto \sqrt[3]{\frac{g \cdot -1}{\mathsf{neg}\left(\frac{0}{a - a}\right)}} \]

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

    13. +-inverses N/A

        \[\leadsto \sqrt[3]{\frac{g \cdot -1}{\mathsf{neg}\left(\frac{1 \cdot 1 - 1 \cdot 1}{0}\right)}} \]

    14. metadata-eval N/A

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

    15. flip-+ N/A

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

    16. metadata-eval N/A

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

    17. metadata-eval N/A

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

    18. associate-*r/ N/A

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

    19. metadata-eval N/A

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

    20. *-commutative N/A

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

    21. lower-cbrt.f64 N/A

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

    22. lift-*.f64 4.5

        \[\leadsto \sqrt[3]{0.5 \cdot g} \]

12. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2025084 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))