2-ancestry mixing, zero discriminant

Percentage Accurate: 75.9% → 98.7%

Time: 6.0s

Alternatives: 10

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: 75.9% 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

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

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 73.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. *-commutative 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

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 73.6%

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

   1. div-inv 73.6%

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

   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 4: 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 73.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 5: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. clear-num 72.9%

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

   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 74.0%

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

   6. metadata-eval 74.0%

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

3. Applied egg-rr 74.0%

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

   1. associate-*r/ 74.0%

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

   2. associate-*l/ 74.0%

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

5. Simplified 74.0%

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

6. Final simplification 74.0%

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

Alternative 6: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 98.7%

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

3. Applied egg-rr 98.7%

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

   1. cbrt-undiv 74.0%

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

   2. *-commutative 74.0%

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

   3. associate-/l* 74.0%

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

5. Applied egg-rr 74.0%

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

6. Final simplification 74.0%

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

Alternative 7: 75.9% 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 73.6%

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

   1. expm1-log1p-u 56.9%

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

   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 39.3%

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

   5. *-un-lft-identity 39.3%

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

   6. times-frac 39.3%

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

   7. metadata-eval 39.3%

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

3. Applied egg-rr 39.3%

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

   1. +-commutative 39.3%

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

   2. associate--l+ 73.6%

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

   3. metadata-eval 73.6%

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

   4. +-rgt-identity 73.6%

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

   5. associate-*r/ 73.6%

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

   6. associate-*l/ 73.6%

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

5. Simplified 73.6%

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

6. Final simplification 73.6%

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

Alternative 8: 17.2% 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 73.6%

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

   1. expm1-log1p-u 56.9%

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

   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 39.3%

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

   5. *-un-lft-identity 39.3%

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

   6. times-frac 39.3%

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

   7. metadata-eval 39.3%

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

3. Applied egg-rr 39.3%

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

   1. +-commutative 39.3%

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

   2. associate--l+ 73.6%

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

   3. metadata-eval 73.6%

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

   4. +-rgt-identity 73.6%

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

   5. associate-*r/ 73.6%

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

   6. associate-*l/ 73.6%

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

5. Simplified 73.6%

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

   1. associate-*l/ 73.6%

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

   2. associate-/l* 72.9%

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

7. Applied egg-rr 72.9%

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

8. Taylor expanded in a around 0 73.6%

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

10. Simplified 16.9%

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

11. Final simplification 16.9%

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

Alternative 9: 6.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-g}{a}}{-2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(g, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: a
    code = (-g / a) / (-2.0d0)
end function

Java

public static double code(double g, double a) {
	return (-g / a) / -2.0;
}

Python

def code(g, a):
	return (-g / a) / -2.0

Julia

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

MATLAB

function tmp = code(g, a)
	tmp = (-g / a) / -2.0;
end

Wolfram

code[g_, a_] := N[(N[((-g) / a), $MachinePrecision] / -2.0), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

   1. clear-num 72.9%

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

   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 74.0%

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

   6. metadata-eval 74.0%

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

3. Applied egg-rr 74.0%

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

   1. associate-*r/ 74.0%

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

   2. associate-/l* 72.8%

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

5. Simplified 72.8%

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

   1. associate-/l* 74.0%

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

   2. cbrt-undiv 98.7%

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

   3. /-rgt-identity 98.7%

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

   4. div-inv 98.7%

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

   5. *-commutative 98.7%

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

   6. associate-/l* 98.7%

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

   7. add-log-exp 4.8%

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

   8. exp-lft-sqr 4.8%

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

   9. log-prod 4.8%

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

   10. add-log-exp 14.3%

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

   11. add-log-exp 98.7%

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

7. Applied egg-rr 98.7%

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

   1. remove-double-div 98.7%

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

   2. clear-num 98.7%

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

   3. count-2 98.7%

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

   4. cbrt-prod 98.2%

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

   5. associate-/l/ 98.1%

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

   6. cbrt-div 73.1%

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

   7. frac-2neg 73.1%

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

   8. distribute-frac-neg 73.1%

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

9. Applied egg-rr 73.1%

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

11. Simplified 6.0%

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

12. Final simplification 6.0%

    \[\leadsto \frac{\frac{-g}{a}}{-2} \]

Alternative 10: 4.0% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(g, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: a
    code = (-2.0d0) / a
end function

Java

public static double code(double g, double a) {
	return -2.0 / a;
}

Python

def code(g, a):
	return -2.0 / a

Julia

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

MATLAB

function tmp = code(g, a)
	tmp = -2.0 / a;
end

Wolfram

code[g_, a_] := N[(-2.0 / a), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

   1. expm1-log1p-u 56.9%

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

   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 39.3%

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

   5. *-un-lft-identity 39.3%

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

   6. times-frac 39.3%

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

   7. metadata-eval 39.3%

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

3. Applied egg-rr 39.3%

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

   1. +-commutative 39.3%

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

   2. associate--l+ 73.6%

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

   3. metadata-eval 73.6%

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

   4. +-rgt-identity 73.6%

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

   5. associate-*r/ 73.6%

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

   6. associate-*l/ 73.6%

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

5. Simplified 73.6%

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

   1. associate-*l/ 73.6%

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

   2. associate-/l* 72.9%

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

7. Applied egg-rr 72.9%

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

   1. clear-num 72.9%

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

   2. associate-/r/ 72.9%

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

   3. clear-num 73.6%

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

   4. metadata-eval 73.6%

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

   5. div-inv 73.6%

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

   6. associate-/l/ 73.6%

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

   7. count-2 73.6%

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

   8. cbrt-undiv 98.7%

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

   9. clear-num 98.7%

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

   10. remove-double-div 98.7%

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

   11. remove-double-div 98.7%

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

9. Applied egg-rr 6.3%

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

10. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0.5 \cdot \left({\left(1 \cdot g\right)}^{0.3333333333333333} \cdot \frac{1}{a}\right)} \]

12. Simplified 4.0%

    \[\leadsto \color{blue}{\frac{-2}{a}} \]

13. Final simplification 4.0%

    \[\leadsto \frac{-2}{a} \]

Reproduce

herbie shell --seed 2023261 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))