2-ancestry mixing, zero discriminant

Percentage Accurate: 76.7% → 98.7%

Time: 7.9s

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: 76.7% 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

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

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 74.4%

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

   1. div-inv 74.4%

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

   2. cbrt-prod 98.8%

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

   3. associate-/r* 98.8%

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

   4. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. clear-num 73.9%

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

   2. cbrt-div 74.9%

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

   3. metadata-eval 74.9%

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

   4. *-un-lft-identity 74.9%

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

   5. times-frac 74.8%

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

   6. metadata-eval 74.8%

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

3. Applied egg-rr 74.8%

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

4. Final simplification 74.8%

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

Alternative 3: 76.6% 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 74.4%

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

   1. clear-num 73.9%

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

   2. cbrt-div 74.9%

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

   3. metadata-eval 74.9%

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

   4. *-un-lft-identity 74.9%

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

   5. times-frac 74.8%

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

   6. metadata-eval 74.8%

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

3. Applied egg-rr 74.8%

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

   1. associate-*r/ 74.9%

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

   2. *-commutative 74.9%

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

   3. add-log-exp 5.3%

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

   4. exp-lft-sqr 5.2%

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

   5. log-prod 5.2%

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

   6. add-log-exp 12.8%

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

   7. add-log-exp 74.9%

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

5. Applied egg-rr 74.9%

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

6. Final simplification 74.9%

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

Alternative 4: 7.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\frac{g}{-8}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= a -5e-310) (cbrt (/ g -8.0)) (cbrt g)))

C

double code(double g, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = cbrt((g / -8.0));
	} else {
		tmp = cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = Math.cbrt((g / -8.0));
	} else {
		tmp = Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (a <= -5e-310)
		tmp = cbrt(Float64(g / -8.0));
	else
		tmp = cbrt(g);
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[a, -5e-310], N[Power[N[(g / -8.0), $MachinePrecision], 1/3], $MachinePrecision], N[Power[g, 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. expm1-log1p-u 57.0%

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

      2. expm1-udef 26.0%

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

      3. log1p-udef 26.0%

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

      4. add-exp-log 45.3%

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

      5. *-un-lft-identity 45.3%

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

      6. times-frac 45.3%

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

      7. metadata-eval 45.3%

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

   3. Applied egg-rr 45.3%

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

      1. +-commutative 45.3%

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

      2. associate--l+ 76.3%

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

      3. metadata-eval 76.3%

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

      4. +-rgt-identity 76.3%

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

      5. associate-*r/ 76.3%

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

      6. associate-*l/ 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. clear-num 76.3%

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

      3. div-inv 76.3%

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

      4. metadata-eval 76.3%

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

      5. div-inv 76.3%

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

      6. associate-/r* 76.3%

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

      7. div-inv 76.3%

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

      8. associate-/l* 76.3%

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

   7. Applied egg-rr 76.3%

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

   9. Applied egg-rr 7.8%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{-8}}} \]

   if -4.999999999999985e-310 < a

   1. Initial program 72.7%

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

      1. clear-num 72.4%

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

      2. cbrt-div 74.0%

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

      3. metadata-eval 74.0%

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

      4. *-un-lft-identity 74.0%

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

      5. times-frac 73.9%

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

      6. metadata-eval 73.9%

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

   3. Applied egg-rr 73.9%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

      3. cbrt-div 98.8%

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

      4. div-inv 98.8%

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

      5. metadata-eval 98.8%

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

      6. div-inv 98.8%

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

      7. add-log-exp 7.1%

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

      8. div-inv 7.1%

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

      9. metadata-eval 7.1%

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

      10. exp-lft-sqr 7.0%

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

      11. log-prod 7.0%

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

      12. add-log-exp 16.9%

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

      13. add-log-exp 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.8%

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

      2. *-rgt-identity 98.8%

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

   7. Simplified 98.8%

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

   9. Applied egg-rr 2.8%

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

   11. Simplified 8.0%

       \[\leadsto \color{blue}{\sqrt[3]{g}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\frac{g}{-8}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g}\\ \end{array} \]

Alternative 5: 7.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\frac{g}{-8}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{0.001953125}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= a -5e-310) (cbrt (/ g -8.0)) (cbrt (/ g 0.001953125))))

C

double code(double g, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = cbrt((g / -8.0));
	} else {
		tmp = cbrt((g / 0.001953125));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = Math.cbrt((g / -8.0));
	} else {
		tmp = Math.cbrt((g / 0.001953125));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (a <= -5e-310)
		tmp = cbrt(Float64(g / -8.0));
	else
		tmp = cbrt(Float64(g / 0.001953125));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[a, -5e-310], N[Power[N[(g / -8.0), $MachinePrecision], 1/3], $MachinePrecision], N[Power[N[(g / 0.001953125), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. expm1-log1p-u 57.0%

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

      2. expm1-udef 26.0%

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

      3. log1p-udef 26.0%

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

      4. add-exp-log 45.3%

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

      5. *-un-lft-identity 45.3%

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

      6. times-frac 45.3%

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

      7. metadata-eval 45.3%

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

   3. Applied egg-rr 45.3%

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

      1. +-commutative 45.3%

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

      2. associate--l+ 76.3%

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

      3. metadata-eval 76.3%

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

      4. +-rgt-identity 76.3%

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

      5. associate-*r/ 76.3%

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

      6. associate-*l/ 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. clear-num 76.3%

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

      3. div-inv 76.3%

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

      4. metadata-eval 76.3%

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

      5. div-inv 76.3%

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

      6. associate-/r* 76.3%

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

      7. div-inv 76.3%

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

      8. associate-/l* 76.3%

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

   7. Applied egg-rr 76.3%

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

   9. Applied egg-rr 7.8%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{-8}}} \]

   if -4.999999999999985e-310 < a

   1. Initial program 72.7%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 22.5%

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

      3. log1p-udef 22.5%

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

      4. add-exp-log 41.7%

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

      5. *-un-lft-identity 41.7%

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

      6. times-frac 41.7%

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

      7. metadata-eval 41.7%

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

   3. Applied egg-rr 41.7%

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

      1. +-commutative 41.7%

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

      2. associate--l+ 72.7%

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

      3. metadata-eval 72.7%

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

      4. +-rgt-identity 72.7%

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

      5. associate-*r/ 72.7%

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

      6. associate-*l/ 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative 72.6%

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

      2. clear-num 72.6%

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

      3. div-inv 72.6%

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

      4. metadata-eval 72.6%

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

      5. div-inv 72.7%

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

      6. associate-/r* 72.7%

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

      7. div-inv 72.6%

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

      8. associate-/l* 72.6%

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

   7. Applied egg-rr 72.6%

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

   9. Applied egg-rr 8.1%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{0.001953125}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\frac{g}{-8}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g}{0.001953125}}\\ \end{array} \]

Alternative 6: 76.7% 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 74.4%

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

   1. expm1-log1p-u 55.1%

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

   2. expm1-udef 24.1%

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

   3. log1p-udef 24.1%

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

   4. add-exp-log 43.4%

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

   5. *-un-lft-identity 43.4%

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

   6. times-frac 43.4%

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

   7. metadata-eval 43.4%

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

3. Applied egg-rr 43.4%

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

   1. +-commutative 43.4%

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

   2. associate--l+ 74.4%

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

   3. metadata-eval 74.4%

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

   4. +-rgt-identity 74.4%

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

   5. associate-*r/ 74.4%

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

   6. associate-*l/ 74.4%

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

5. Simplified 74.4%

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

6. Final simplification 74.4%

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

Alternative 7: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

2. Final simplification 74.4%

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

Alternative 8: 7.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (if (<= a -5e-310) (- (cbrt g)) (cbrt g)))

C

double code(double g, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = -cbrt(g);
	} else {
		tmp = cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (a <= -5e-310) {
		tmp = -Math.cbrt(g);
	} else {
		tmp = Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (a <= -5e-310)
		tmp = Float64(-cbrt(g));
	else
		tmp = cbrt(g);
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[a, -5e-310], (-N[Power[g, 1/3], $MachinePrecision]), N[Power[g, 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. clear-num 75.6%

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

      2. cbrt-div 75.9%

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

      3. metadata-eval 75.9%

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

      4. *-un-lft-identity 75.9%

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

      5. times-frac 75.9%

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

      6. metadata-eval 75.9%

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

   3. Applied egg-rr 75.9%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. cbrt-div 98.7%

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

      4. div-inv 98.6%

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

      5. metadata-eval 98.6%

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

      6. div-inv 98.6%

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

      7. add-log-exp 3.2%

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

      8. div-inv 3.2%

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

      9. metadata-eval 3.2%

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

      10. exp-lft-sqr 3.2%

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

      11. log-prod 3.2%

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

      12. add-log-exp 11.7%

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

      13. add-log-exp 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.7%

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

      2. *-rgt-identity 98.7%

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

   7. Simplified 98.7%

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

   9. Applied egg-rr 1.6%

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

   11. Simplified 7.8%

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

   if -4.999999999999985e-310 < a

   1. Initial program 72.7%

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

      1. clear-num 72.4%

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

      2. cbrt-div 74.0%

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

      3. metadata-eval 74.0%

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

      4. *-un-lft-identity 74.0%

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

      5. times-frac 73.9%

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

      6. metadata-eval 73.9%

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

   3. Applied egg-rr 73.9%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

      3. cbrt-div 98.8%

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

      4. div-inv 98.8%

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

      5. metadata-eval 98.8%

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

      6. div-inv 98.8%

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

      7. add-log-exp 7.1%

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

      8. div-inv 7.1%

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

      9. metadata-eval 7.1%

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

      10. exp-lft-sqr 7.0%

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

      11. log-prod 7.0%

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

      12. add-log-exp 16.9%

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

      13. add-log-exp 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.8%

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

      2. *-rgt-identity 98.8%

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

   7. Simplified 98.8%

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

   9. Applied egg-rr 2.8%

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

   11. Simplified 8.0%

       \[\leadsto \color{blue}{\sqrt[3]{g}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g}\\ \end{array} \]

Alternative 9: 4.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. clear-num 73.9%

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

   2. cbrt-div 74.9%

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

   3. metadata-eval 74.9%

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

   4. *-un-lft-identity 74.9%

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

   5. times-frac 74.8%

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

   6. metadata-eval 74.8%

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

3. Applied egg-rr 74.8%

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

   1. associate-*r/ 74.9%

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

   2. *-commutative 74.9%

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

   3. cbrt-div 98.8%

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

   4. div-inv 98.7%

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

   5. metadata-eval 98.7%

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

   6. div-inv 98.7%

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

   7. add-log-exp 5.3%

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

   8. div-inv 5.3%

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

   9. metadata-eval 5.3%

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

   10. exp-lft-sqr 5.2%

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

   11. log-prod 5.2%

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

   12. add-log-exp 14.4%

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

   13. add-log-exp 98.7%

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

5. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.8%

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

   2. *-rgt-identity 98.8%

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

7. Simplified 98.8%

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

9. Applied egg-rr 2.7%

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

11. Simplified 5.0%

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

12. Final simplification 5.0%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))