2-ancestry mixing, zero discriminant

Percentage Accurate: 75.7% → 98.7%

Time: 4.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

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

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

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, 0.6× speedup

Math

\[\sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g + g} \]

FPCore

(FPCore (g a)
  :precision binary64
  (* (cbrt (/ 0.25 a)) (cbrt (+ g g))))

C

double code(double g, double a) {
	return cbrt((0.25 / a)) * cbrt((g + g));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((0.25 / a)) * Math.cbrt((g + g));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(0.25 / a)) * cbrt(Float64(g + g)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(0.25 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7%

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

3. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip-rev N/A

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

   5. div-flip N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound-/.f64 98.7%

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g + g}} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7%

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

3. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}}} \]
4. Add Preprocessing

Alternative 3: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7%

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

3. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip-rev N/A

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

   5. div-flip N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound-/.f64 98.7%

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

5. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-flip-rev N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. cbrt-undiv N/A

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

   9. cbrt-neg-rev N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. associate-/r* N/A

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

   13. lift-cbrt.f64 N/A

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

   14. cbrt-prod N/A

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

   15. lift-*.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. mult-flip-rev N/A

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

   18. lift-/.f64 N/A

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

   19. lift-neg.f64 N/A

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

   20. distribute-lft-neg-out N/A

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

   21. distribute-rgt-neg-in N/A

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

   22. lift-/.f64 N/A

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

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g}} \]
8. Add Preprocessing

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\sqrt[3]{g} \cdot \frac{0.7937005259840998}{\sqrt[3]{a}} \]

FPCore

(FPCore (g a)
  :precision binary64
  (* (cbrt g) (/ 0.7937005259840998 (cbrt a))))

C

double code(double g, double a) {
	return cbrt(g) * (0.7937005259840998 / cbrt(a));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(g) * (0.7937005259840998 / Math.cbrt(a));
}

Julia

function code(g, a)
	return Float64(cbrt(g) * Float64(0.7937005259840998 / cbrt(a)))
end

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] * N[(0.7937005259840998 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7%

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

3. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. lift-cbrt.f64 N/A

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

   6. associate-/l* N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. cbrt-div N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\color{blue}{\frac{2}{4}}} \]

   13. cbrt-div N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{4}}} \]

   14. lift-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{\sqrt[3]{2}}{\color{blue}{\sqrt[3]{4}}} \]

   15. associate-/l* N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{2}}{\sqrt[3]{4}}} \]

   16. cbrt-prod N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{a} \cdot 2}}}{\sqrt[3]{4}} \]

   17. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   19. lift-cbrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   20. mult-flip N/A

       \[\leadsto \color{blue}{\sqrt[3]{2 \cdot \frac{g}{a}} \cdot \frac{1}{\sqrt[3]{4}}} \]

5. Applied rewrites 75.7%

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

6. Evaluated real constant 75.7%

   \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{0.7937005259840998} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \frac{7149018786131517}{9007199254740992}} \]

   2. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \cdot \frac{7149018786131517}{9007199254740992} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a}}} \cdot \frac{7149018786131517}{9007199254740992} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a}}} \cdot \frac{7149018786131517}{9007199254740992} \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{g}}}{\sqrt[3]{a}} \cdot \frac{7149018786131517}{9007199254740992} \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \frac{7149018786131517}{9007199254740992}}{\sqrt[3]{a}}} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \frac{\frac{7149018786131517}{9007199254740992}}{\sqrt[3]{a}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \frac{\frac{7149018786131517}{9007199254740992}}{\sqrt[3]{a}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\frac{\frac{7149018786131517}{9007199254740992}}{\sqrt[3]{a}}} \]

   10. lower-cbrt.f64 98.6%

       \[\leadsto \sqrt[3]{g} \cdot \frac{0.7937005259840998}{\color{blue}{\sqrt[3]{a}}} \]

8. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \frac{0.7937005259840998}{\sqrt[3]{a}}} \]
9. Add Preprocessing

Alternative 5: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.6%

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

   7. lift-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 98.6%

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

3. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a + a}}} \]
4. Add Preprocessing

Alternative 6: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-105}:\\ \;\;\;\;e^{\left(\log \left(\left|g\right| \cdot 0.5\right) - \log \left(\left|a\right|\right)\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_0 \leq 10^{+102}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{\left|a\right|} \cdot \left|g\right|}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(\left|g\right|\right) - \log \left(\left|a\right| + \left|a\right|\right)\right) \cdot 0.3333333333333333}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
  :precision binary64
  (let* ((t_0 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
  (*
   (copysign 1.0 g)
   (*
    (copysign 1.0 a)
    (if (<= t_0 2e-105)
      (exp
       (*
        (- (log (* (fabs g) 0.5)) (log (fabs a)))
        0.3333333333333333))
      (if (<= t_0 1e+102)
        (cbrt (* (/ 0.5 (fabs a)) (fabs g)))
        (exp
         (*
          (- (log (fabs g)) (log (+ (fabs a) (fabs a))))
          0.3333333333333333))))))))

C

double code(double g, double a) {
	double t_0 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_0 <= 2e-105) {
		tmp = exp(((log((fabs(g) * 0.5)) - log(fabs(a))) * 0.3333333333333333));
	} else if (t_0 <= 1e+102) {
		tmp = cbrt(((0.5 / fabs(a)) * fabs(g)));
	} else {
		tmp = exp(((log(fabs(g)) - log((fabs(a) + fabs(a)))) * 0.3333333333333333));
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_0 <= 2e-105) {
		tmp = Math.exp(((Math.log((Math.abs(g) * 0.5)) - Math.log(Math.abs(a))) * 0.3333333333333333));
	} else if (t_0 <= 1e+102) {
		tmp = Math.cbrt(((0.5 / Math.abs(a)) * Math.abs(g)));
	} else {
		tmp = Math.exp(((Math.log(Math.abs(g)) - Math.log((Math.abs(a) + Math.abs(a)))) * 0.3333333333333333));
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_0 <= 2e-105)
		tmp = exp(Float64(Float64(log(Float64(abs(g) * 0.5)) - log(abs(a))) * 0.3333333333333333));
	elseif (t_0 <= 1e+102)
		tmp = cbrt(Float64(Float64(0.5 / abs(a)) * abs(g)));
	else
		tmp = exp(Float64(Float64(log(abs(g)) - log(Float64(abs(a) + abs(a)))) * 0.3333333333333333));
	end
	return Float64(copysign(1.0, g) * Float64(copysign(1.0, a) * tmp))
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(N[Abs[g], $MachinePrecision] / N[(2.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[g]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 2e-105], N[Exp[N[(N[(N[Log[N[(N[Abs[g], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] - N[Log[N[Abs[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+102], N[Power[N[(N[(0.5 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[g], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] - N[Log[N[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1.9999999999999999e-105

   1. Initial program 75.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-log.f64 35.2%

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 35.2%

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-/r* N/A

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

      6. log-div N/A

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

      7. mult-flip-rev N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-unsound--.f64 N/A

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

      12. lower-unsound-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-unsound-log.f64 22.4%

          \[\leadsto e^{\left(\log \left(g \cdot 0.5\right) - \color{blue}{\log a}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.4%

      \[\leadsto e^{\color{blue}{\left(\log \left(g \cdot 0.5\right) - \log a\right)} \cdot 0.3333333333333333} \]

   if 1.9999999999999999e-105 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999998e101

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.7%

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

   3. Applied rewrites 75.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.8%

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

   5. Applied rewrites 75.8%

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

   if 9.9999999999999998e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 75.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-log.f64 35.2%

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 35.2%

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-log.f64 N/A

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

      6. lower-unsound-log.f64 22.4%

         \[\leadsto e^{\left(\log g - \color{blue}{\log \left(a + a\right)}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.4%

      \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{\left(\log \left(\left|g\right|\right) - \log \left(\left|a\right| + \left|a\right|\right)\right) \cdot 0.3333333333333333}\\ t_1 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+102}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{\left|a\right|} \cdot \left|g\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
  :precision binary64
  (let* ((t_0
        (exp
         (*
          (- (log (fabs g)) (log (+ (fabs a) (fabs a))))
          0.3333333333333333)))
       (t_1 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
  (*
   (copysign 1.0 g)
   (*
    (copysign 1.0 a)
    (if (<= t_1 2e-105)
      t_0
      (if (<= t_1 1e+102)
        (cbrt (* (/ 0.5 (fabs a)) (fabs g)))
        t_0))))))

C

double code(double g, double a) {
	double t_0 = exp(((log(fabs(g)) - log((fabs(a) + fabs(a)))) * 0.3333333333333333));
	double t_1 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_1 <= 2e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+102) {
		tmp = cbrt(((0.5 / fabs(a)) * fabs(g)));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Java

public static double code(double g, double a) {
	double t_0 = Math.exp(((Math.log(Math.abs(g)) - Math.log((Math.abs(a) + Math.abs(a)))) * 0.3333333333333333));
	double t_1 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_1 <= 2e-105) {
		tmp = t_0;
	} else if (t_1 <= 1e+102) {
		tmp = Math.cbrt(((0.5 / Math.abs(a)) * Math.abs(g)));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = exp(Float64(Float64(log(abs(g)) - log(Float64(abs(a) + abs(a)))) * 0.3333333333333333))
	t_1 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_1 <= 2e-105)
		tmp = t_0;
	elseif (t_1 <= 1e+102)
		tmp = cbrt(Float64(Float64(0.5 / abs(a)) * abs(g)));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, g) * Float64(copysign(1.0, a) * tmp))
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] - N[Log[N[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Abs[g], $MachinePrecision] / N[(2.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[g]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 2e-105], t$95$0, If[LessEqual[t$95$1, 1e+102], N[Power[N[(N[(0.5 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[g], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1.9999999999999999e-105 or 9.9999999999999998e101 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 75.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-log.f64 35.2%

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 35.2%

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-log.f64 N/A

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

      6. lower-unsound-log.f64 22.4%

         \[\leadsto e^{\left(\log g - \color{blue}{\log \left(a + a\right)}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.4%

      \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot 0.3333333333333333} \]

   if 1.9999999999999999e-105 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999998e101

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.7%

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

   3. Applied rewrites 75.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.8%

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

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a} \cdot g}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.8% accurate, 1.0× speedup

Math

\[\sqrt[3]{\frac{g}{a}} \cdot 0.7937005259840998 \]

FPCore

(FPCore (g a)
  :precision binary64
  (* (cbrt (/ g a)) 0.7937005259840998))

C

double code(double g, double a) {
	return cbrt((g / a)) * 0.7937005259840998;
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / a)) * 0.7937005259840998;
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(g / a)) * 0.7937005259840998)
end

Wolfram

code[g_, a_] := N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] * 0.7937005259840998), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7%

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

3. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. lift-cbrt.f64 N/A

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

   6. associate-/l* N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. cbrt-div N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\color{blue}{\frac{2}{4}}} \]

   13. cbrt-div N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{4}}} \]

   14. lift-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{\sqrt[3]{2}}{\color{blue}{\sqrt[3]{4}}} \]

   15. associate-/l* N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{2}}{\sqrt[3]{4}}} \]

   16. cbrt-prod N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{a} \cdot 2}}}{\sqrt[3]{4}} \]

   17. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   19. lift-cbrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   20. mult-flip N/A

       \[\leadsto \color{blue}{\sqrt[3]{2 \cdot \frac{g}{a}} \cdot \frac{1}{\sqrt[3]{4}}} \]

5. Applied rewrites 75.7%

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

6. Evaluated real constant 75.7%

   \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{0.7937005259840998} \]
7. Add Preprocessing

Alternative 9: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 75.7%

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

3. Applied rewrites 75.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 75.8%

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

5. Applied rewrites 75.8%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a} \cdot g}} \]
6. Add Preprocessing

Alternative 10: 75.7% accurate, 1.0× speedup

Math

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

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 75.7%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 75.7%

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

3. Applied rewrites 75.7%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))