2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 2.1s

Alternatives: 3

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: 76.4% 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

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

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 74.9%

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

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

   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 2: 76.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. flip3-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{{a}^{3} + {a}^{3}}{\color{blue}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}}} \]

   9. lower--.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{{a}^{3} + {a}^{3}}{\mathsf{fma}\left(a, a, \color{blue}{a \cdot a - a \cdot a}\right)}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt[3]{\frac{g}{\frac{{a}^{3} + {a}^{3}}{\mathsf{fma}\left(a, a, \color{blue}{a \cdot a} - a \cdot a\right)}}} \]

   11. lower-*.f64 27.1

       \[\leadsto \sqrt[3]{\frac{g}{\frac{{a}^{3} + {a}^{3}}{\mathsf{fma}\left(a, a, a \cdot a - \color{blue}{a \cdot a}\right)}}} \]

4. Applied rewrites 27.1%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\frac{{a}^{3} + {a}^{3}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{{a}^{3} + {a}^{3}}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{{a}^{3}} + {a}^{3}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   3. lift-pow.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{{a}^{3} + \color{blue}{{a}^{3}}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   4. sum-cubes N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\left(a \cdot a + \left(a \cdot a - a \cdot a\right)\right) \cdot \left(a + a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\left(a \cdot a + \left(\color{blue}{a \cdot a} - a \cdot a\right)\right) \cdot \left(a + a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\left(a \cdot a + \left(a \cdot a - \color{blue}{a \cdot a}\right)\right) \cdot \left(a + a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   7. lift--.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\left(a \cdot a + \color{blue}{\left(a \cdot a - a \cdot a\right)}\right) \cdot \left(a + a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)} \cdot \left(a + a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right) \cdot \left(a + a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   10. lift--.f64 N/A

       \[\leadsto \sqrt[3]{\frac{g}{\frac{\mathsf{fma}\left(a, a, \color{blue}{a \cdot a - a \cdot a}\right) \cdot \left(a + a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   11. +-inverses N/A

       \[\leadsto \sqrt[3]{\frac{g}{\frac{\mathsf{fma}\left(a, a, \color{blue}{0}\right) \cdot \left(a + a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   12. count-2-rev N/A

       \[\leadsto \sqrt[3]{\frac{g}{\frac{\mathsf{fma}\left(a, a, 0\right) \cdot \color{blue}{\left(2 \cdot a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   13. lower-*.f64 27.1

       \[\leadsto \sqrt[3]{\frac{g}{\frac{\mathsf{fma}\left(a, a, 0\right) \cdot \color{blue}{\left(2 \cdot a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

6. Applied rewrites 27.1%

   \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \left(2 \cdot a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\frac{\mathsf{fma}\left(a, a, 0\right) \cdot \left(2 \cdot a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \left(2 \cdot a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   3. lift-fma.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\left(a \cdot a + 0\right)} \cdot \left(2 \cdot a\right)}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\frac{\left(a \cdot a + 0\right) \cdot \color{blue}{\left(2 \cdot a\right)}}{\mathsf{fma}\left(a, a, a \cdot a - a \cdot a\right)}}} \]

   5. +-rgt-identity N/A

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

   6. lift-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. +-inverses N/A

       \[\leadsto \sqrt[3]{\frac{g}{\left(a \cdot a\right) \cdot \frac{2 \cdot a}{a \cdot a + \color{blue}{0}}}} \]

   14. +-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-*.f64 40.8

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

8. Applied rewrites 40.8%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\left(a \cdot a\right) \cdot \frac{a \cdot 2}{a \cdot a}}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 41.9

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 74.9

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

10. Applied rewrites 74.9%

    \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a \cdot \left(a \cdot \left(\frac{a}{a} \cdot \frac{2}{a}\right)\right)}}} \]
11. Add Preprocessing

Alternative 3: 76.4% 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 74.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 74.9

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

4. Applied rewrites 74.9%

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

Reproduce

herbie shell --seed 2025057 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))