2-ancestry mixing, zero discriminant

Percentage Accurate: 76.5% → 98.7%

Time: 3.0s

Alternatives: 8

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: 76.5% 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

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

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 76.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 2: 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 76.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 3: 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 76.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 4: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\left|a\right|}\\ t_1 := e^{\mathsf{fma}\left(\log t\_0, 0.3333333333333333, \log \left(\left|g\right|\right) \cdot 0.3333333333333333\right)}\\ t_2 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{t\_0 \cdot \left|g\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ 0.5 (fabs a)))
        (t_1
         (exp
          (fma
           (log t_0)
           0.3333333333333333
           (* (log (fabs g)) 0.3333333333333333))))
        (t_2 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
   (*
    (copysign 1.0 g)
    (*
     (copysign 1.0 a)
     (if (<= t_2 1e-105)
       t_1
       (if (<= t_2 4e+100) (cbrt (* t_0 (fabs g))) t_1))))))

C

double code(double g, double a) {
	double t_0 = 0.5 / fabs(a);
	double t_1 = exp(fma(log(t_0), 0.3333333333333333, (log(fabs(g)) * 0.3333333333333333)));
	double t_2 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_2 <= 1e-105) {
		tmp = t_1;
	} else if (t_2 <= 4e+100) {
		tmp = cbrt((t_0 * fabs(g)));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = Float64(0.5 / abs(a))
	t_1 = exp(fma(log(t_0), 0.3333333333333333, Float64(log(abs(g)) * 0.3333333333333333)))
	t_2 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_2 <= 1e-105)
		tmp = t_1;
	elseif (t_2 <= 4e+100)
		tmp = cbrt(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[(0.5 / N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[Log[t$95$0], $MachinePrecision] * 0.3333333333333333 + N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $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, 1e-105], t$95$1, If[LessEqual[t$95$2, 4e+100], N[Power[N[(t$95$0 * N[Abs[g], $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))) < 9.9999999999999997e-106 or 4.0000000000000001e100 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0%

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

      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 36.0%

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.5%

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

   5. Applied rewrites 22.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. neg-log N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. count-2 N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 22.5%

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

   7. Applied rewrites 22.5%

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

   if 9.9999999999999997e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.0000000000000001e100

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 76.5%

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

   3. Applied rewrites 76.5%

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

Alternative 5: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\left|g\right|\right)\\ t_1 := \frac{0.5}{\left|a\right|}\\ t_2 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-105}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, 0.3333333333333333, \log t\_1 \cdot 0.3333333333333333\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{t\_1 \cdot \left|g\right|}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(t\_0 - \log \left(\left|a\right| + \left|a\right|\right)\right) \cdot 0.3333333333333333}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (log (fabs g)))
        (t_1 (/ 0.5 (fabs a)))
        (t_2 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
   (*
    (copysign 1.0 g)
    (*
     (copysign 1.0 a)
     (if (<= t_2 1e-105)
       (exp (fma t_0 0.3333333333333333 (* (log t_1) 0.3333333333333333)))
       (if (<= t_2 4e+100)
         (cbrt (* t_1 (fabs g)))
         (exp (* (- t_0 (log (+ (fabs a) (fabs a)))) 0.3333333333333333))))))))

C

double code(double g, double a) {
	double t_0 = log(fabs(g));
	double t_1 = 0.5 / fabs(a);
	double t_2 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_2 <= 1e-105) {
		tmp = exp(fma(t_0, 0.3333333333333333, (log(t_1) * 0.3333333333333333)));
	} else if (t_2 <= 4e+100) {
		tmp = cbrt((t_1 * fabs(g)));
	} else {
		tmp = exp(((t_0 - log((fabs(a) + fabs(a)))) * 0.3333333333333333));
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = log(abs(g))
	t_1 = Float64(0.5 / abs(a))
	t_2 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_2 <= 1e-105)
		tmp = exp(fma(t_0, 0.3333333333333333, Float64(log(t_1) * 0.3333333333333333)));
	elseif (t_2 <= 4e+100)
		tmp = cbrt(Float64(t_1 * abs(g)));
	else
		tmp = exp(Float64(Float64(t_0 - log(Float64(abs(a) + abs(a)))) * 0.3333333333333333));
	end
	return Float64(copysign(1.0, g) * Float64(copysign(1.0, a) * tmp))
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Abs[a], $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, 1e-105], N[Exp[N[(t$95$0 * 0.3333333333333333 + N[(N[Log[t$95$1], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+100], N[Power[N[(t$95$1 * N[Abs[g], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(t$95$0 - N[Log[N[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999997e-106

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0%

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

      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 36.0%

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.5%

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

   5. Applied rewrites 22.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. neg-log N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

      13. count-2 N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-/.f64 22.5%

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

   7. Applied rewrites 22.5%

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

   if 9.9999999999999997e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.0000000000000001e100

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 76.5%

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

   3. Applied rewrites 76.5%

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

   if 4.0000000000000001e100 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0%

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

      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 36.0%

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.5%

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

   5. Applied rewrites 22.5%

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

Alternative 6: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{\left(\log \left(\left|g\right|\right) - \log \left(\left|a\right| + \left|a\right|\right)\right) \cdot 0.3333333333333333}\\ t_1 := \sqrt[3]{\frac{\left|g\right|}{2 \cdot \left|a\right|}}\\ \mathsf{copysign}\left(1, g\right) \cdot \left(\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{\left|a\right|} \cdot \left|g\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0
         (exp
          (*
           (- (log (fabs g)) (log (+ (fabs a) (fabs a))))
           0.3333333333333333)))
        (t_1 (cbrt (/ (fabs g) (* 2.0 (fabs a))))))
   (*
    (copysign 1.0 g)
    (*
     (copysign 1.0 a)
     (if (<= t_1 1e-105)
       t_0
       (if (<= t_1 4e+100) (cbrt (* (/ 0.5 (fabs a)) (fabs g))) t_0))))))

C

double code(double g, double a) {
	double t_0 = exp(((log(fabs(g)) - log((fabs(a) + fabs(a)))) * 0.3333333333333333));
	double t_1 = cbrt((fabs(g) / (2.0 * fabs(a))));
	double tmp;
	if (t_1 <= 1e-105) {
		tmp = t_0;
	} else if (t_1 <= 4e+100) {
		tmp = cbrt(((0.5 / fabs(a)) * fabs(g)));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, g) * (copysign(1.0, a) * tmp);
}

Java

public static double code(double g, double a) {
	double t_0 = Math.exp(((Math.log(Math.abs(g)) - Math.log((Math.abs(a) + Math.abs(a)))) * 0.3333333333333333));
	double t_1 = Math.cbrt((Math.abs(g) / (2.0 * Math.abs(a))));
	double tmp;
	if (t_1 <= 1e-105) {
		tmp = t_0;
	} else if (t_1 <= 4e+100) {
		tmp = Math.cbrt(((0.5 / Math.abs(a)) * Math.abs(g)));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, g) * (Math.copySign(1.0, a) * tmp);
}

Julia

function code(g, a)
	t_0 = exp(Float64(Float64(log(abs(g)) - log(Float64(abs(a) + abs(a)))) * 0.3333333333333333))
	t_1 = cbrt(Float64(abs(g) / Float64(2.0 * abs(a))))
	tmp = 0.0
	if (t_1 <= 1e-105)
		tmp = t_0;
	elseif (t_1 <= 4e+100)
		tmp = cbrt(Float64(Float64(0.5 / abs(a)) * abs(g)));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, g) * Float64(copysign(1.0, a) * tmp))
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Abs[g], $MachinePrecision]], $MachinePrecision] - N[Log[N[(N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Abs[g], $MachinePrecision] / N[(2.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[g]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 1e-105], t$95$0, If[LessEqual[t$95$1, 4e+100], N[Power[N[(N[(0.5 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[g], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 9.9999999999999997e-106 or 4.0000000000000001e100 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.0%

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

      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 36.0%

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

   3. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   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 22.5%

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

   5. Applied rewrites 22.5%

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

   if 9.9999999999999997e-106 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 4.0000000000000001e100

   1. Initial program 76.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 76.5%

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

   3. Applied rewrites 76.5%

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

Alternative 7: 76.5% 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 76.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 76.5%

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

3. Applied rewrites 76.5%

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

Alternative 8: 76.5% 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 76.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 76.5%

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

3. Applied rewrites 76.5%

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

Reproduce

herbie shell --seed 2025189 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))