2-ancestry mixing, zero discriminant

Percentage Accurate: 75.7% → 98.6%

Time: 5.1s

Alternatives: 5

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 75.7% 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: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{1}{\sqrt[3]{g}}}{\sqrt[3]{\frac{0.5}{a}}}} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (/ 1.0 (/ (/ 1.0 (cbrt g)) (cbrt (/ 0.5 a)))))

C

double code(double g, double a) {
	return 1.0 / ((1.0 / cbrt(g)) / cbrt((0.5 / a)));
}

Java

public static double code(double g, double a) {
	return 1.0 / ((1.0 / Math.cbrt(g)) / Math.cbrt((0.5 / a)));
}

Julia

function code(g, a)
	return Float64(1.0 / Float64(Float64(1.0 / cbrt(g)) / cbrt(Float64(0.5 / a))))
end

Wolfram

code[g_, a_] := N[(1.0 / N[(N[(1.0 / N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. clear-num 73.2%

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

   2. cbrt-div 73.6%

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

   3. metadata-eval 73.6%

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

   4. *-un-lft-identity 73.6%

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

   5. times-frac 73.6%

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

   6. metadata-eval 73.6%

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

3. Applied egg-rr 73.6%

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

   1. associate-*r/ 73.6%

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

   2. associate-*l/ 73.2%

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

5. Simplified 73.2%

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

   1. associate-*l/ 73.6%

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

   2. cbrt-undiv 98.7%

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

   3. clear-num 98.7%

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

   4. div-inv 98.7%

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

   5. associate-/r* 98.6%

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

   6. *-commutative 98.6%

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

   7. metadata-eval 98.6%

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

   8. metadata-eval 98.6%

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

   9. div-inv 98.6%

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

   10. cbrt-div 98.8%

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

   11. clear-num 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. div-inv 74.2%

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

   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: 5.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g \cdot \left(a + a\right)} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. div-inv 74.2%

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

   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. Step-by-step derivation

   1. clear-num 98.7%

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

   2. cbrt-div 98.7%

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

   3. metadata-eval 98.7%

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

   4. div-inv 98.7%

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

   5. metadata-eval 98.7%

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

   6. div-inv 98.7%

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

   7. clear-num 98.7%

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

   8. *-commutative 98.7%

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

   9. div-inv 98.7%

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

   10. associate-/r* 98.7%

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

   11. *-commutative 98.7%

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

   12. metadata-eval 98.7%

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

   13. metadata-eval 98.7%

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

   14. div-inv 98.7%

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

   15. cbrt-div 98.7%

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

   16. clear-num 98.7%

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

5. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{0.5}{a}}}{\frac{1}{\sqrt[3]{g}}}} \]
6. Step-by-step derivation

   1. associate-/r/ 98.7%

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

   2. /-rgt-identity 98.7%

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

   3. clear-num 98.7%

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

   4. cbrt-div 98.7%

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

   5. metadata-eval 98.7%

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

   6. div-inv 98.7%

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

   7. metadata-eval 98.7%

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

   8. metadata-eval 98.7%

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

   9. distribute-rgt-out 98.7%

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

   10. *-un-lft-identity 98.7%

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

   11. *-un-lft-identity 98.7%

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

   12. add-cbrt-cube 98.5%

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

   13. cbrt-unprod 73.9%

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

7. Applied egg-rr 5.8%

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

8. Final simplification 5.8%

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

Alternative 4: 75.7% 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 74.2%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 24.5%

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

   3. log1p-udef 24.5%

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

   4. add-exp-log 43.7%

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

   5. *-un-lft-identity 43.7%

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

   6. times-frac 43.7%

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

   7. metadata-eval 43.7%

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

3. Applied egg-rr 43.7%

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

   1. +-commutative 43.7%

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

   2. associate--l+ 74.2%

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

   3. metadata-eval 74.2%

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

   4. +-rgt-identity 74.2%

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

   5. associate-*r/ 74.2%

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

   6. associate-*l/ 74.2%

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

5. Simplified 74.2%

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

6. Final simplification 74.2%

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

Alternative 5: 75.7% 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 74.2%

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

2. Final simplification 74.2%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))