2-ancestry mixing, zero discriminant

Percentage Accurate: 75.4% → 98.6%

Time: 6.1s

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: 75.4% 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: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. cbrt-div 98.7%

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

   2. clear-num 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

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 78.6%

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

   1. div-inv 78.6%

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

   2. cbrt-prod 98.7%

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

   3. associate-/r* 98.7%

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

   4. metadata-eval 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 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 78.6%

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

   1. cbrt-div 98.7%

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

   2. div-inv 98.7%

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

3. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.7%

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

   2. *-rgt-identity 98.7%

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

   3. *-commutative 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. clear-num 78.0%

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

   2. cbrt-div 78.7%

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

   3. metadata-eval 78.7%

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

   4. *-un-lft-identity 78.7%

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

   5. times-frac 78.6%

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

   6. metadata-eval 78.6%

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

3. Applied egg-rr 78.6%

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

   1. associate-*r/ 78.7%

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

   2. associate-*l/ 78.4%

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

5. Simplified 78.4%

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

   1. associate-*l/ 78.7%

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

   2. add-log-exp 4.0%

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

   3. *-commutative 4.0%

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

   4. exp-lft-sqr 3.9%

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

   5. log-prod 3.9%

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

   6. add-log-exp 10.9%

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

   7. add-log-exp 78.7%

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

7. Applied egg-rr 78.7%

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

8. Final simplification 78.7%

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

Alternative 5: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. expm1-log1p-u 56.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 23.9%

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

   4. add-exp-log 46.2%

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

   5. *-un-lft-identity 46.2%

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

   6. times-frac 46.2%

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

   7. metadata-eval 46.2%

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

3. Applied egg-rr 46.2%

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

   1. +-commutative 46.2%

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

   2. associate--l+ 78.6%

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

   3. metadata-eval 78.6%

      \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} + \color{blue}{0} \]

   4. +-rgt-identity 78.6%

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

   5. associate-*r/ 78.6%

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

   6. associate-*l/ 78.6%

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

5. Simplified 78.6%

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

6. Final simplification 78.6%

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

Alternative 6: 75.4% 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]

Derivation

1. Initial program 78.6%

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

2. Final simplification 78.6%

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

Alternative 7: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. expm1-log1p-u 56.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 23.9%

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

   4. add-exp-log 46.2%

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

   5. *-un-lft-identity 46.2%

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

   6. times-frac 46.2%

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

   7. metadata-eval 46.2%

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

3. Applied egg-rr 46.2%

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

   1. +-commutative 46.2%

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

   2. associate--l+ 78.6%

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

   3. metadata-eval 78.6%

      \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} + \color{blue}{0} \]

   4. +-rgt-identity 78.6%

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

   5. associate-*r/ 78.6%

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

   6. associate-*l/ 78.6%

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

5. Simplified 78.6%

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

   1. *-commutative 78.6%

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

   2. associate-*r/ 78.6%

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

7. Applied egg-rr 78.6%

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

8. Final simplification 78.6%

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

Alternative 8: 17.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. cbrt-div 98.7%

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

   2. clear-num 98.8%

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

3. Applied egg-rr 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/l* 98.7%

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

   3. add-cube-cbrt 97.7%

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

   4. times-frac 97.7%

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

5. Applied egg-rr 97.7%

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

   1. times-frac 97.7%

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

   2. *-lft-identity 97.7%

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

   3. unpow2 97.7%

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

   4. rem-3cbrt-lft 98.7%

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

7. Simplified 98.7%

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

   1. flip-+ 0.0%

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

   2. difference-of-squares 0.0%

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

   3. +-inverses 0.0%

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

   4. metadata-eval 0.0%

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

   5. +-inverses 0.0%

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\frac{\left(a + a\right) \cdot \log 1}{\color{blue}{0}}}} \]

   6. metadata-eval 0.0%

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

   7. associate-*r/ 0.0%

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

   8. metadata-eval 0.0%

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\left(a + a\right) \cdot \frac{\color{blue}{0}}{\log 1}}} \]

   9. +-inverses 0.0%

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

   10. difference-of-squares 0.0%

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

   11. +-inverses 0.0%

       \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\left(a + a\right) \cdot \frac{\left(a + a\right) \cdot \color{blue}{0}}{\log 1}}} \]

   12. metadata-eval 0.0%

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

   13. associate-*r/ 0.0%

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

   14. metadata-eval 0.0%

       \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\left(a + a\right) \cdot \left(\left(a + a\right) \cdot \frac{\color{blue}{0}}{\log 1}\right)}} \]

   15. +-inverses 0.0%

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

   16. metadata-eval 0.0%

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

   17. +-inverses 0.0%

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

   18. flip-+ 5.4%

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

   19. associate-*l* 5.4%

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

   20. add-cbrt-cube 6.4%

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

   21. add-cube-cbrt 6.4%

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

   22. add-cube-cbrt 6.4%

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

9. Applied egg-rr 6.4%

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

11. Simplified 21.3%

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

    1. expm1-log1p-u 15.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right)\right)} \]

    2. expm1-udef 8.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right)} - 1} \]

    3. cbrt-undiv 7.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt[3]{\frac{g}{a}}}\right)} - 1 \]

13. Applied egg-rr 7.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 12.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)\right)} \]

    2. expm1-log1p 17.7%

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

15. Simplified 17.7%

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

16. Final simplification 17.7%

    \[\leadsto \sqrt[3]{\frac{g}{a}} \]

Alternative 9: 4.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-2} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt -2.0))

C

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

Java

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

Julia

function code(g, a)
	return cbrt(-2.0)
end

Wolfram

code[g_, a_] := N[Power[-2.0, 1/3], $MachinePrecision]

Derivation

1. Initial program 78.6%

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

   1. expm1-log1p-u 56.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 23.9%

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

   4. add-exp-log 46.2%

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

   5. *-un-lft-identity 46.2%

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

   6. times-frac 46.2%

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

   7. metadata-eval 46.2%

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

3. Applied egg-rr 46.2%

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

   1. +-commutative 46.2%

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

   2. associate--l+ 78.6%

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

   3. metadata-eval 78.6%

      \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} + \color{blue}{0} \]

   4. +-rgt-identity 78.6%

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

   5. associate-*r/ 78.6%

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

   6. associate-*l/ 78.6%

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

5. Simplified 78.6%

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

   1. associate-*l/ 78.6%

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

   2. associate-/l* 78.0%

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

7. Applied egg-rr 78.0%

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

8. Taylor expanded in a around 0 78.6%

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

10. Simplified 4.6%

    \[\leadsto \sqrt[3]{\color{blue}{-2}} \]

11. Final simplification 4.6%

    \[\leadsto \sqrt[3]{-2} \]

Reproduce

herbie shell --seed 2023222 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))