2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 3.4s

Alternatives: 9

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: 76.4% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. lift-cbrt.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. mult-flip N/A

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

   9. frac-times N/A

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

   10. cbrt-div N/A

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

   11. lower-/.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-*.f64 98.6%

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

   16. lift-*.f64 N/A

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

   17. count-2-rev N/A

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

   18. lower-+.f64 98.6%

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

3. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 98.5%

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 98.5%

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

   11. lift-*.f64 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 98.5%

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. cbrt-prod N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. count-2 N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 98.6%

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

7. Applied rewrites 98.6%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\frac{\sqrt[3]{g + g}}{\sqrt[3]{4 \cdot a}} \]

FPCore

(FPCore (g a)
  :precision binary64
  (/ (cbrt (+ g g)) (cbrt (* 4.0 a))))

C

double code(double g, double a) {
	return cbrt((g + g)) / cbrt((4.0 * a));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g + g)) / Math.cbrt((4.0 * a));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(g + g)) / cbrt(Float64(4.0 * a)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(4.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

   1. lift-cbrt.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. mult-flip N/A

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

   9. frac-times N/A

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

   10. cbrt-div N/A

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

   11. lower-/.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-*.f64 98.6%

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

   16. lift-*.f64 N/A

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

   17. count-2-rev N/A

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

   18. lower-+.f64 98.6%

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

3. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 98.6%

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 98.6%

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

5. Applied rewrites 98.6%

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

Alternative 3: 98.6% 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 76.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 4: 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 76.4%

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

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

   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 98.7%

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

3. Applied rewrites 98.7%

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

Alternative 5: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|a\right|\\ t_1 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-104}:\\ \;\;\;\;e^{\left(\log \left(\frac{1}{\left|a\right|}\right) + \log \left(\left|g\right| \cdot 0.5\right)\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_1 \leq 10^{+99}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{t\_0}{\left|g\right|}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(\left|g\right|\right) - \log t\_0\right) \cdot 0.3333333333333333}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs a)))
       (t_1 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
  (*
   (copysign 1.0 g)
   (*
    (copysign 1.0 a)
    (if (<= t_1 5e-104)
      (exp
       (*
        (+ (log (/ 1.0 (fabs a))) (log (* (fabs g) 0.5)))
        0.3333333333333333))
      (if (<= t_1 1e+99)
        (cbrt (/ 1.0 (/ t_0 (fabs g))))
        (exp (* (- (log (fabs g)) (log t_0)) 0.3333333333333333))))))))

C

double code(double g, double a) {
	double t_0 = fabs(a) + fabs(a);
	double t_1 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_1 <= 5e-104) {
		tmp = exp(((log((1.0 / fabs(a))) + log((fabs(g) * 0.5))) * 0.3333333333333333));
	} else if (t_1 <= 1e+99) {
		tmp = cbrt((1.0 / (t_0 / fabs(g))));
	} else {
		tmp = exp(((log(fabs(g)) - log(t_0)) * 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.abs(a) + Math.abs(a);
	double t_1 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_1 <= 5e-104) {
		tmp = Math.exp(((Math.log((1.0 / Math.abs(a))) + Math.log((Math.abs(g) * 0.5))) * 0.3333333333333333));
	} else if (t_1 <= 1e+99) {
		tmp = Math.cbrt((1.0 / (t_0 / Math.abs(g))));
	} else {
		tmp = Math.exp(((Math.log(Math.abs(g)) - Math.log(t_0)) * 0.3333333333333333));
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = Float64(abs(a) + abs(a))
	t_1 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_1 <= 5e-104)
		tmp = exp(Float64(Float64(log(Float64(1.0 / abs(a))) + log(Float64(abs(g) * 0.5))) * 0.3333333333333333));
	elseif (t_1 <= 1e+99)
		tmp = cbrt(Float64(1.0 / Float64(t_0 / abs(g))));
	else
		tmp = exp(Float64(Float64(log(abs(g)) - log(t_0)) * 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[(N[Abs[a], $MachinePrecision] + N[Abs[a], $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, 5e-104], N[Exp[N[(N[(N[Log[N[(1.0 / N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[N[(N[Abs[g], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+99], N[Power[N[(1.0 / N[(t$95$0 / N[Abs[g], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] - N[Log[t$95$0], $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))) < 4.9999999999999998e-104

   1. Initial program 76.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.1%

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

      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 36.1%

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

   3. Applied rewrites 36.1%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. log-prod N/A

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

      13. lower-unsound-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. count-2 N/A

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

      16. associate-/l/ N/A

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

      17. metadata-eval N/A

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

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

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

      19. lower-/.f64 N/A

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

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

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

      21. mult-flip N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 22.4%

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

   5. Applied rewrites 22.4%

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

   if 4.9999999999999998e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999997e98

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 75.7%

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 75.7%

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

   3. Applied rewrites 75.7%

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

   if 9.9999999999999997e98 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.1%

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

      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 36.1%

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

   3. Applied rewrites 36.1%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 6: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|a\right|\\ t_1 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-104}:\\ \;\;\;\;e^{\left(\log \left(\left|g\right| \cdot 0.5\right) - \log \left(\left|a\right|\right)\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;t\_1 \leq 10^{+99}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{t\_0}{\left|g\right|}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(\left|g\right|\right) - \log t\_0\right) \cdot 0.3333333333333333}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs a)))
       (t_1 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
  (*
   (copysign 1.0 g)
   (*
    (copysign 1.0 a)
    (if (<= t_1 5e-104)
      (exp
       (*
        (- (log (* (fabs g) 0.5)) (log (fabs a)))
        0.3333333333333333))
      (if (<= t_1 1e+99)
        (cbrt (/ 1.0 (/ t_0 (fabs g))))
        (exp (* (- (log (fabs g)) (log t_0)) 0.3333333333333333))))))))

C

double code(double g, double a) {
	double t_0 = fabs(a) + fabs(a);
	double t_1 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_1 <= 5e-104) {
		tmp = exp(((log((fabs(g) * 0.5)) - log(fabs(a))) * 0.3333333333333333));
	} else if (t_1 <= 1e+99) {
		tmp = cbrt((1.0 / (t_0 / fabs(g))));
	} else {
		tmp = exp(((log(fabs(g)) - log(t_0)) * 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.abs(a) + Math.abs(a);
	double t_1 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_1 <= 5e-104) {
		tmp = Math.exp(((Math.log((Math.abs(g) * 0.5)) - Math.log(Math.abs(a))) * 0.3333333333333333));
	} else if (t_1 <= 1e+99) {
		tmp = Math.cbrt((1.0 / (t_0 / Math.abs(g))));
	} else {
		tmp = Math.exp(((Math.log(Math.abs(g)) - Math.log(t_0)) * 0.3333333333333333));
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = Float64(abs(a) + abs(a))
	t_1 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_1 <= 5e-104)
		tmp = exp(Float64(Float64(log(Float64(abs(g) * 0.5)) - log(abs(a))) * 0.3333333333333333));
	elseif (t_1 <= 1e+99)
		tmp = cbrt(Float64(1.0 / Float64(t_0 / abs(g))));
	else
		tmp = exp(Float64(Float64(log(abs(g)) - log(t_0)) * 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[(N[Abs[a], $MachinePrecision] + N[Abs[a], $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, 5e-104], 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$1, 1e+99], N[Power[N[(1.0 / N[(t$95$0 / N[Abs[g], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] - N[Log[t$95$0], $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))) < 4.9999999999999998e-104

   1. Initial program 76.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.1%

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

      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 36.1%

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

   3. Applied rewrites 36.1%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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. lower-unsound--.f64 N/A

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

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

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. 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 4.9999999999999998e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999997e98

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 75.7%

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 75.7%

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

   3. Applied rewrites 75.7%

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

   if 9.9999999999999997e98 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.1%

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

      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 36.1%

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

   3. Applied rewrites 36.1%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|a\right|\\ t_1 := e^{\left(\log \left(\left|g\right|\right) - \log t\_0\right) \cdot 0.3333333333333333}\\ t_2 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+99}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{t\_0}{\left|g\right|}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs a)))
       (t_1 (exp (* (- (log (fabs g)) (log t_0)) 0.3333333333333333)))
       (t_2 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
  (*
   (copysign 1.0 g)
   (*
    (copysign 1.0 a)
    (if (<= t_2 5e-104)
      t_1
      (if (<= t_2 1e+99) (cbrt (/ 1.0 (/ t_0 (fabs g)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = fabs(a) + fabs(a);
	double t_1 = exp(((log(fabs(g)) - log(t_0)) * 0.3333333333333333));
	double t_2 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_2 <= 5e-104) {
		tmp = t_1;
	} else if (t_2 <= 1e+99) {
		tmp = cbrt((1.0 / (t_0 / fabs(g))));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Java

public static double code(double g, double a) {
	double t_0 = Math.abs(a) + Math.abs(a);
	double t_1 = Math.exp(((Math.log(Math.abs(g)) - Math.log(t_0)) * 0.3333333333333333));
	double t_2 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_2 <= 5e-104) {
		tmp = t_1;
	} else if (t_2 <= 1e+99) {
		tmp = Math.cbrt((1.0 / (t_0 / Math.abs(g))));
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = Float64(abs(a) + abs(a))
	t_1 = exp(Float64(Float64(log(abs(g)) - log(t_0)) * 0.3333333333333333))
	t_2 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_2 <= 5e-104)
		tmp = t_1;
	elseif (t_2 <= 1e+99)
		tmp = cbrt(Float64(1.0 / Float64(t_0 / abs(g))));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, g) * Float64(copysign(1.0, a) * tmp))
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] - N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$2, 5e-104], t$95$1, If[LessEqual[t$95$2, 1e+99], N[Power[N[(1.0 / N[(t$95$0 / N[Abs[g], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.9999999999999998e-104 or 9.9999999999999997e98 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.1%

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

      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 36.1%

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

   3. Applied rewrites 36.1%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 4.9999999999999998e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999997e98

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 75.7%

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 75.7%

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

   3. Applied rewrites 75.7%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a + a}{g}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.4% accurate, 1.0× speedup

Math

\[\frac{\sqrt[3]{\frac{g}{a}}}{1.2599210498948732} \]

FPCore

(FPCore (g a)
  :precision binary64
  (/ (cbrt (/ g a)) 1.2599210498948732))

C

double code(double g, double a) {
	return cbrt((g / a)) / 1.2599210498948732;
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / a)) / 1.2599210498948732;
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(g / a)) / 1.2599210498948732)
end

Wolfram

code[g_, a_] := N[(N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision] / 1.2599210498948732), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. cbrt-prod N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. cbrt-undiv N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-cbrt.f64 75.7%

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

3. Applied rewrites 75.7%

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

4. Evaluated real constant 76.4%

   \[\leadsto \frac{\sqrt[3]{\frac{g}{a}}}{\color{blue}{1.2599210498948732}} \]
5. Add Preprocessing

Alternative 9: 76.4% 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 76.4%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 76.4%

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

3. Applied rewrites 76.4%

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

Reproduce

herbie shell --seed 2025210 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))