2-ancestry mixing, zero discriminant

Percentage Accurate: 75.7% → 98.7%

Time: 4.1s

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: 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

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

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 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] / N[(1.2599210498948732 * 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. 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 75.0

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

3. Applied rewrites 75.0%

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

4. Evaluated real constant 75.7%

   \[\leadsto \frac{\sqrt[3]{\frac{g}{a}}}{\color{blue}{\frac{5674179970822795}{4503599627370496}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{a}}}{\frac{5674179970822795}{4503599627370496}}} \]

   2. lift-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{a}}}}{\frac{5674179970822795}{4503599627370496}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{a}}}}{\frac{5674179970822795}{4503599627370496}} \]

   4. cbrt-div N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a}}}}{\frac{5674179970822795}{4503599627370496}} \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{g}}}{\sqrt[3]{a}}}{\frac{5674179970822795}{4503599627370496}} \]

   6. lift-cbrt.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}}}{\frac{5674179970822795}{4503599627370496}} \]

   7. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot \frac{5674179970822795}{4503599627370496}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a} \cdot \frac{5674179970822795}{4503599627370496}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\frac{5674179970822795}{4503599627370496} \cdot \sqrt[3]{a}}} \]

   10. lower-*.f64 98.7

       \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{1.2599210498948732 \cdot \sqrt[3]{a}}} \]

6. Applied rewrites 98.7%

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

Alternative 3: 98.7% 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. mult-flip N/A

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

   4. *-commutative N/A

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

   5. cbrt-prod N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7

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

3. Applied rewrites 98.7%

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

Alternative 4: 98.7% 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 98.7

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

   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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{\left|g\right|}\\ t_1 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-106}:\\ \;\;\;\;e^{\left(\log t\_0 + \log \left(\left|a\right|\right)\right) \cdot -0.3333333333333333}\\ \mathbf{elif}\;t\_1 \leq 10^{+103}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{\left|a\right|}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(\left|a\right| + \left|a\right|\right) - \log \left(\left|g\right|\right)\right) \cdot -0.3333333333333333}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ 2.0 (fabs g))) (t_1 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
   (*
    (copysign 1.0 g)
    (*
     (copysign 1.0 a)
     (if (<= t_1 4e-106)
       (exp (* (+ (log t_0) (log (fabs a))) -0.3333333333333333))
       (if (<= t_1 1e+103)
         (cbrt (/ (/ 1.0 (fabs a)) t_0))
         (exp
          (*
           (- (log (+ (fabs a) (fabs a))) (log (fabs g)))
           -0.3333333333333333))))))))

C

double code(double g, double a) {
	double t_0 = 2.0 / fabs(g);
	double t_1 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_1 <= 4e-106) {
		tmp = exp(((log(t_0) + log(fabs(a))) * -0.3333333333333333));
	} else if (t_1 <= 1e+103) {
		tmp = cbrt(((1.0 / fabs(a)) / t_0));
	} else {
		tmp = exp(((log((fabs(a) + fabs(a))) - log(fabs(g))) * -0.3333333333333333));
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Java

public static double code(double g, double a) {
	double t_0 = 2.0 / Math.abs(g);
	double t_1 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_1 <= 4e-106) {
		tmp = Math.exp(((Math.log(t_0) + Math.log(Math.abs(a))) * -0.3333333333333333));
	} else if (t_1 <= 1e+103) {
		tmp = Math.cbrt(((1.0 / Math.abs(a)) / t_0));
	} else {
		tmp = Math.exp(((Math.log((Math.abs(a) + Math.abs(a))) - Math.log(Math.abs(g))) * -0.3333333333333333));
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = Float64(2.0 / abs(g))
	t_1 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_1 <= 4e-106)
		tmp = exp(Float64(Float64(log(t_0) + log(abs(a))) * -0.3333333333333333));
	elseif (t_1 <= 1e+103)
		tmp = cbrt(Float64(Float64(1.0 / abs(a)) / t_0));
	else
		tmp = exp(Float64(Float64(log(Float64(abs(a) + abs(a))) - log(abs(g))) * -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[(2.0 / N[Abs[g], $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, 4e-106], N[Exp[N[(N[(N[Log[t$95$0], $MachinePrecision] + N[Log[N[Abs[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+103], N[Power[N[(N[(1.0 / N[Abs[a], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(N[Log[N[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[Abs[g], $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))) < 3.99999999999999976e-106

   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. *-commutative N/A

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

      5. cbrt-prod N/A

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

      6. metadata-eval N/A

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

      7. cbrt-undiv N/A

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

      8. metadata-eval N/A

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

      9. lift-cbrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-cbrt.f64 N/A

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

      12. cbrt-div N/A

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

      13. lift-/.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. mult-flip N/A

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

      16. lift-cbrt.f64 N/A

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

      17. lift-cbrt.f64 N/A

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

      18. cbrt-undiv N/A

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

      19. div-flip-rev N/A

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

      20. lift-/.f64 N/A

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

      21. inv-pow N/A

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

      22. cbrt-pow N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. pow-to-exp N/A

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

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

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

   5. Applied rewrites 35.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. log-prod N/A

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

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

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

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

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

      10. lower-unsound-log.f64 22.8

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

   7. Applied rewrites 22.8%

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

   if 3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1e103

   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%

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

      1. lift-/.f64 N/A

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

      2. div-flip-rev N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if 1e103 < (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. 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. *-commutative N/A

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

      5. cbrt-prod N/A

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

      6. metadata-eval N/A

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

      7. cbrt-undiv N/A

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

      8. metadata-eval N/A

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

      9. lift-cbrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-cbrt.f64 N/A

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

      12. cbrt-div N/A

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

      13. lift-/.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. mult-flip N/A

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

      16. lift-cbrt.f64 N/A

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

      17. lift-cbrt.f64 N/A

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

      18. cbrt-undiv N/A

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

      19. div-flip-rev N/A

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

      20. lift-/.f64 N/A

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

      21. inv-pow N/A

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

      22. cbrt-pow N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. pow-to-exp N/A

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

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

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

   5. Applied rewrites 35.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

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

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

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

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

      6. lower-unsound-log.f64 22.8

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

   7. Applied rewrites 22.8%

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

Alternative 6: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{\left(\log \left(\left|a\right| + \left|a\right|\right) - \log \left(\left|g\right|\right)\right) \cdot -0.3333333333333333}\\ t_1 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+103}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{\left|a\right|}}{\frac{2}{\left|g\right|}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0
         (exp
          (*
           (- (log (+ (fabs a) (fabs a))) (log (fabs g)))
           -0.3333333333333333)))
        (t_1 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
   (*
    (copysign 1.0 g)
    (*
     (copysign 1.0 a)
     (if (<= t_1 4e-106)
       t_0
       (if (<= t_1 1e+103)
         (cbrt (/ (/ 1.0 (fabs a)) (/ 2.0 (fabs g))))
         t_0))))))

C

double code(double g, double a) {
	double t_0 = exp(((log((fabs(a) + fabs(a))) - log(fabs(g))) * -0.3333333333333333));
	double t_1 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_1 <= 4e-106) {
		tmp = t_0;
	} else if (t_1 <= 1e+103) {
		tmp = cbrt(((1.0 / fabs(a)) / (2.0 / 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(a) + Math.abs(a))) - Math.log(Math.abs(g))) * -0.3333333333333333));
	double t_1 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_1 <= 4e-106) {
		tmp = t_0;
	} else if (t_1 <= 1e+103) {
		tmp = Math.cbrt(((1.0 / Math.abs(a)) / (2.0 / 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(Float64(abs(a) + abs(a))) - log(abs(g))) * -0.3333333333333333))
	t_1 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_1 <= 4e-106)
		tmp = t_0;
	elseif (t_1 <= 1e+103)
		tmp = cbrt(Float64(Float64(1.0 / abs(a)) / Float64(2.0 / 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[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[Abs[g], $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, 4e-106], t$95$0, If[LessEqual[t$95$1, 1e+103], N[Power[N[(N[(1.0 / N[Abs[a], $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[Abs[g], $MachinePrecision]), $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))) < 3.99999999999999976e-106 or 1e103 < (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. 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. *-commutative N/A

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

      5. cbrt-prod N/A

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

      6. metadata-eval N/A

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

      7. cbrt-undiv N/A

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

      8. metadata-eval N/A

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

      9. lift-cbrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-cbrt.f64 N/A

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

      12. cbrt-div N/A

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

      13. lift-/.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. mult-flip N/A

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

      16. lift-cbrt.f64 N/A

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

      17. lift-cbrt.f64 N/A

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

      18. cbrt-undiv N/A

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

      19. div-flip-rev N/A

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

      20. lift-/.f64 N/A

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

      21. inv-pow N/A

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

      22. cbrt-pow N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. pow-to-exp N/A

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

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

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

   5. Applied rewrites 35.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

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

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

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

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

      6. lower-unsound-log.f64 22.8

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

   7. Applied rewrites 22.8%

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

   if 3.99999999999999976e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1e103

   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%

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

      1. lift-/.f64 N/A

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

      2. div-flip-rev N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

Alternative 7: 75.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, a_] := N[(1.0 / N[Power[N[(N[(a + a), $MachinePrecision] / 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

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

   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. *-commutative N/A

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

   5. cbrt-prod N/A

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

   6. metadata-eval N/A

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

   7. cbrt-undiv N/A

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

   8. metadata-eval N/A

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

   9. lift-cbrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lift-cbrt.f64 N/A

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

   12. cbrt-div N/A

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

   13. lift-/.f64 N/A

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

   14. lift-cbrt.f64 N/A

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

   15. mult-flip N/A

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

   16. div-flip-rev N/A

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

   17. metadata-eval N/A

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

   18. lift-cbrt.f64 N/A

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

   19. lift-cbrt.f64 N/A

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

   20. cbrt-div N/A

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

   21. lift-/.f64 N/A

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

   22. metadata-eval N/A

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

   23. lower-/.f64 N/A

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

   24. lower-cbrt.f64 75.0

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

5. Applied rewrites 75.7%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a + a}{g}}}} \]
6. Add Preprocessing

Alternative 8: 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

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 75.7

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

3. Applied rewrites 75.7%

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

Alternative 9: 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%

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

Reproduce

herbie shell --seed 2025171 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))