2-ancestry mixing, zero discriminant

Percentage Accurate: 77.1% → 98.7%

Time: 3.1s

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: 77.1% 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 75.6%

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

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

   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. 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 98.8

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

4. Applied rewrites 98.8%

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

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 98.8

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

7. Applied rewrites 98.8%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

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

   3. frac-2neg N/A

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

   4. cbrt-div N/A

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

   5. lift-*.f64 N/A

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

   6. count-2-rev N/A

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

   7. flip-+ N/A

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

   8. distribute-neg-frac N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. +-inverses N/A

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

   12. flip-+ N/A

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

   13. count-2-rev N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cbrt.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-cbrt.f64 1.6

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. flip-+ N/A

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

   22. +-inverses N/A

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{\frac{\color{blue}{0}}{a - a}}} \]

   23. metadata-eval N/A

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

   24. +-inverses N/A

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

   25. distribute-neg-frac N/A

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{-g}}{\sqrt[3]{-2 \cdot a}}} \]
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. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. cbrt-div N/A

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

   8. lift-neg.f64 N/A

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

   9. frac-2neg N/A

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

   10. cbrt-div N/A

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

   11. lift-cbrt.f64 N/A

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

   12. cbrt-prod N/A

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

   13. lift-cbrt.f64 N/A

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

   14. pow1/3 N/A

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

   15. exp-to-pow N/A

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

   16. lift-log.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-exp.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-cbrt.f64 N/A

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

   21. lift-exp.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. lift-log.f64 N/A

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

   24. exp-to-pow N/A

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

6. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]
7. 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 75.6%

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

3. Taylor expanded in g around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 75.6

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

5. Applied rewrites 75.6%

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

7. Applied rewrites 39.1%

   \[\leadsto {\left(\frac{g}{a} \cdot 0.5\right)}^{\color{blue}{0.3333333333333333}} \]
8. Step-by-step derivation

9. Applied rewrites 98.6%

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

Alternative 4: 77.1% 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 75.6%

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

3. Taylor expanded in g around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 75.6

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

5. Applied rewrites 75.6%

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

7. Applied rewrites 39.1%

   \[\leadsto {\left(\frac{g}{a} \cdot 0.5\right)}^{\color{blue}{0.3333333333333333}} \]
8. Step-by-step derivation

9. Applied rewrites 98.6%

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

11. Applied rewrites 75.7%

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

Alternative 5: 77.1% 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 75.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 75.6

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

4. Applied rewrites 75.6%

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

Reproduce

herbie shell --seed 2024346 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))