2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.6%

Time: 6.6s

Alternatives: 5

Speedup: 0.5×

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: 76.3% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. associate-/r* 75.1%

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

   2. cbrt-div 98.8%

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

   3. div-inv 98.8%

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

   4. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. clear-num 98.7%

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

   2. inv-pow 98.7%

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

   3. div-inv 98.7%

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

   4. metadata-eval 98.7%

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

   5. metadata-eval 98.7%

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

   6. div-inv 98.7%

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

   7. cbrt-div 98.7%

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

   8. clear-num 98.7%

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

   9. unpow-prod-down 98.7%

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

   10. inv-pow 98.7%

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

6. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a}} \cdot {\left(\sqrt[3]{\frac{2}{g}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 98.7%

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

   2. associate-*r/ 98.7%

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

   3. *-rgt-identity 98.7%

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

8. Simplified 98.7%

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

   1. cbrt-div 98.1%

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

   2. associate-/r/ 98.1%

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

   3. associate-/r* 98.1%

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

   4. cbrt-prod 98.4%

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

   5. associate-/r/ 98.4%

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

   6. clear-num 98.4%

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

   7. cbrt-undiv 74.8%

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

   8. *-commutative 74.8%

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

   9. *-un-lft-identity 74.8%

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

   10. times-frac 75.1%

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

   11. metadata-eval 75.1%

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

   12. clear-num 74.4%

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

   13. div-inv 74.4%

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

   14. cbrt-div 74.8%

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

   15. cbrt-div 98.8%

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

   16. associate-/r/ 98.7%

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

10. Applied egg-rr 98.7%

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

    1. associate-*l/ 98.8%

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

    2. associate-/l* 98.8%

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

12. Simplified 98.8%

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

13. Final simplification 98.8%

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

Alternative 2: 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 74.8%

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

   1. pow1/3 37.2%

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

   2. clear-num 36.8%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{2 \cdot a}{g}}\right)}}^{0.3333333333333333} \]

   3. associate-/r/ 37.2%

      \[\leadsto {\color{blue}{\left(\frac{1}{2 \cdot a} \cdot g\right)}}^{0.3333333333333333} \]

   4. unpow-prod-down 27.6%

      \[\leadsto \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{0.3333333333333333} \cdot {g}^{0.3333333333333333}} \]

   5. pow1/3 47.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \cdot {g}^{0.3333333333333333} \]

   6. associate-/r* 48.3%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}}} \cdot {g}^{0.3333333333333333} \]

   7. metadata-eval 48.3%

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

   8. pow1/3 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 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 74.8%

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

   1. associate-/r* 75.1%

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

   2. cbrt-div 98.8%

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

   3. div-inv 98.8%

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

   4. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ (/ 1.0 a) (/ 2.0 g))))

C

double code(double g, double a) {
	return cbrt(((1.0 / a) / (2.0 / g)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(((1.0 / a) / (2.0 / g)));
}

Julia

function code(g, a)
	return cbrt(Float64(Float64(1.0 / a) / Float64(2.0 / g)))
end

Wolfram

code[g_, a_] := N[Power[N[(N[(1.0 / a), $MachinePrecision] / N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 74.8%

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

   1. associate-/r* 75.1%

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

   2. cbrt-div 98.8%

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

   3. div-inv 98.8%

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

   4. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. clear-num 98.7%

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

   2. inv-pow 98.7%

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

   3. div-inv 98.7%

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

   4. metadata-eval 98.7%

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

   5. metadata-eval 98.7%

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

   6. div-inv 98.7%

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

   7. cbrt-div 98.7%

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

   8. clear-num 98.7%

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

   9. unpow-prod-down 98.7%

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

   10. inv-pow 98.7%

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

6. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a}} \cdot {\left(\sqrt[3]{\frac{2}{g}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 98.7%

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

   2. associate-*r/ 98.7%

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

   3. *-rgt-identity 98.7%

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

8. Simplified 98.7%

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

   1. add-cbrt-cube 98.4%

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

   2. cbrt-undiv 74.8%

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

   3. un-div-inv 74.8%

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

   4. frac-times 74.8%

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

   5. metadata-eval 74.8%

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

   6. associate-/r* 74.8%

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

   7. add-cube-cbrt 75.1%

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

10. Applied egg-rr 75.1%

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

11. Final simplification 75.1%

    \[\leadsto \sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}} \]
12. Add Preprocessing

Alternative 5: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in g around 0 75.1%

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

4. Final simplification 75.1%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))