2-ancestry mixing, zero discriminant

Percentage Accurate: 76.2% → 98.7%

Time: 7.7s

Alternatives: 4

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: 76.2% 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 79.6%

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

   1. div-inv 79.7%

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

   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: 76.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. cbrt-div 98.8%

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

   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/ 98.8%

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

   2. *-rgt-identity 98.8%

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

   3. *-commutative 98.8%

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

5. Simplified 98.8%

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

   1. cbrt-div 79.6%

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

   2. clear-num 78.6%

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

   3. associate-/r/ 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-/r* 79.7%

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

   6. metadata-eval 79.7%

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

   7. associate-/r/ 78.6%

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

   8. div-inv 78.6%

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

   9. cbrt-prod 78.7%

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

   10. clear-num 79.8%

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

7. Applied egg-rr 79.8%

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

8. Final simplification 79.8%

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

Alternative 3: 76.2% 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 79.6%

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

   1. expm1-log1p-u 60.6%

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

   2. expm1-udef 26.2%

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

   3. log1p-udef 26.2%

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

   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+ 79.6%

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

   3. metadata-eval 79.6%

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

   4. +-rgt-identity 79.6%

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

   5. associate-*r/ 79.6%

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

   6. associate-*l/ 79.7%

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

5. Simplified 79.7%

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

   1. add-cube-cbrt 79.5%

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

   2. cbrt-prod 79.5%

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

   3. cbrt-prod 79.4%

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

   4. *-commutative 79.4%

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

   5. *-commutative 79.4%

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

   6. cbrt-prod 79.3%

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

   7. *-commutative 79.3%

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

   8. pow3 79.4%

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

   9. cbrt-unprod 79.5%

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

   10. clear-num 79.5%

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

   11. un-div-inv 79.5%

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

   12. div-inv 79.5%

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

   13. metadata-eval 79.5%

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

7. Applied egg-rr 79.5%

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

   1. unpow3 79.5%

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

   2. add-cube-cbrt 79.6%

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

   3. clear-num 78.6%

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

   4. associate-*r/ 78.6%

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

   5. associate-/r* 79.7%

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

9. Applied egg-rr 79.7%

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

10. Final simplification 79.7%

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

Alternative 4: 76.3% 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 79.6%

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

   1. expm1-log1p-u 60.6%

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

   2. expm1-udef 26.2%

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

   3. log1p-udef 26.2%

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

   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+ 79.6%

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

   3. metadata-eval 79.6%

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

   4. +-rgt-identity 79.6%

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

   5. associate-*r/ 79.6%

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

   6. associate-*l/ 79.7%

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

5. Simplified 79.7%

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

6. Final simplification 79.7%

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

Reproduce

herbie shell --seed 2023282 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))